Machlokes whether a non-kohen may place both hands on the head of someone he is bentching.
This Machlokes follows from the opinions of Rashi and the Ramban in Parshas Shemini regarding the first time Aharon blessed the people, whether he used the language of Birkas Kohanim in Naso or some other nusach.
Whether Birkas Kohanim is De'oraysa or Derabanan in our time.
Why Birkas Kohanim is after Modim and why we say Ve'se'erav.
The concept of Birkas Kohanim following Avodas Korbanos and the extent to which tefilla is like a korban.
There is a widespread minhag to bless children Friday night. This minhag is brought in the siddur of Reb Yaakov Emden and in the siddur of the Gr'a. In Rav Emden's siddur (section on ליל שבת), he says that one should place both hands on the child during the Bracha. He says that those who believe that one should use only one hand are incorrect. Or, as he puts it, ולא כחסירי דעת שחושבין שיש קפידא לברך דוקא ביד אחת. (If you've ever read his Mor Uketzi'ah, you know that in matters of Halacha and Hashkafa, Rav Emden was not a model of tranquility, and "חסירי דעת" is, by his standards, mild.) This opinion is also stated in the Pachad Yitzchak (Reb Yitzchak Lampronti, section on Brachos, page 54, who says that although some only use one hand, he uses both. (He doesn't say that they are חסירי דעת.)
On the other hand, the Siddur Hagr'a (section on ליל שבת) says that the Gaon is quoted as having said that it is assur for anyone other than a Kohen to give a bracha with both hands.
The Torah Temima here also quotes the Gaon to this effect, and says that when the Vilner Posek Rav Landau got married, he got a bracha from the Gaon, who put one hand on him. He asked the Gaon why one hand, and the Gaon said that we do not find a two handed bracha other than kohanim in the Mikdash. Irrespective of your opinion on Rabbi Epstein's reliability, this particular story has been said by others with a significant difference. There is a book of stories of gedolim called Ma'asei Ilfas by a man named Ilfas, published in Jerusalem in תש"א. The book was published when the author was an elderly man, and he says (page 9) that he personally met the Vilner Dayan (who died in תרל"א at 91), and he himself told him the story. When he was a Chassan, he came in to the Gaon's house while the Gaon was eating kugel at the day meal of Shabbos, and the Gaon put out his hands to give him a bracha. I am not making this up. The young rabbi Landau moved back, because he was afraid the Gaon would make his new Shtreimel greasy. So the Gaon ended up putting only one hand on him and giving him the bracha. Rabbi Landau said that his whole life he regretted the childish preference for a clean shtreimel over the Gaon's hand.
The stories in the Torah Temima and the Ma'aseh Ilfas cancel each other out. But we do have the Siddur Hagr'a that validates the story of the Gaon's k'peida to use only one hand, and Reb Moshe once told me that the authors and editors of the Siddur Hagr'a were talmidei chachamim who took their work very seriously and can be trusted as to factual matters.
The basis for the Gaon's k'peida, one assumes, is the Gemara (Ksuvos 24b) that a non-Kohen that lifts his hands and says birkas Kohanim transgresses the mitzva of our parsha, namely, דַּבֵּר אֶל אַהֲרֹן וְאֶל בָּנָיו לֵאמֹר כֹּה תְבָרְכוּ אֶת בְּנֵי יִשְׂרָאֵל . The passuk is telling us that only Aharon and his children can use this protocol of Bracha.
Exactly when the issur applies is well discussed: the Magen Avraham (beginning of OC 128) says it's only a problem when using the Sheim Hashem, either in a bracha of Levarech or in the words of the psukim of Birkas Kohanim (see Noda BeYehuda OC I:5). The Ksav Sofer says the issur is only when intending to fulfill the Mitzva of Koh Sevarchu. The Bach there says the issur is only when one raises his hands, which is consistent with the Gaon.
I would say that the issue of whether you can use both hands when duchenning depends on Rashi/Ramban in Shmini, and the Mishneh Le'Melech.
In Parshas Shemini (Vayikra 9:22) it says that after inaugurating the Mizbei'ach, Aharon raised his hands and blessed the people-וַיִּשָּׂא אַהֲרֹן אֶת יָדָו אֶל הָעָם וַיְבָרְכֵם. Rashi there says that (even though the parsha of Birkas Kohanim had not yet been taught) Aharon was inspired in his choice of words and he gave the blessing exactly as it was later taught in our parsha- Yevarech'echa Hashem Veyishme'recha.... The Ramban disagrees and says that Aharon's bracha was just a bracha, and we have no idea what the words of the bracha were, just as Shlomo Hamelech gave the people a bracha when the Beis Hamikdash was finished. He says that the Bracha in our parsha was created only when Hashem taught it to Moshe.
The Mishneh Le'Melech brings this machlokes in Hilchos Tefilla (14:9). He explains that although we learn many dinim of Birkas Kohanim from Parshas Shmini, this doesn't prove like Rashi. The limudim are only because of a Gzeira Shava, and don't prove that the two brachos are identical. The Ramban, therefore, learns that the bracha of Shmini was entirely different, and the Gzeira Shava only imports the circumstances of Shmini and applies them to the new parsha of Birkas Kohanim. Rashi, on the other hand, holds that the two brachos are the same, and that we would know many dinim without a Gzeira Shava, because Shmini is also Birkas Kohanim. According to Rashi, we only need the Gzeira Shava to teach us that you don't need to match the Bracha of Shmini exactly, that it doesn't have to be the Kohen Gadol on Rosh Chodesh.
So: according to Rashi, Shemini's Vayisa Kapav is specifically and exclusively associated with Birkas Kohanim, and a non-kohen that uses both hands is doing something that is intended to be unique to Duchening. According to the Ramban, on the other hand, the raised hands of Shemini are not a unique characteristic of duchenning, because Aharon wasn't duchenning when he raised his hands. That action is appropriate to brachos in general. When you give someone a bracha, you reach out to him, you want contact. If so, there is no reason to think that "אֶל אַהֲרֹן וְאֶל בָּנָיו לֵאמֹר כֹּה תְבָרְכוּ" precludes others from using both hands when they give a bracha. Only those things that are uniquely associated with duchenning are prohibited to non-Kohanim.
The Gemara (Sotah 38a) uses the passuk in Shmini to tell us that Birkas Kohanim can only be done when the Kohen finishes an Avoda of Korbanos Tzibur.
Because of this, Reb Yaakov Emden (same place as above, and also in his Mor Uketziah 128) says that Birkas Kohanim in our time is Miderabanan. Mid'oraysa, no Avodah means no Duchenning. The Mishna Berura (Shaar Tziyun in 128) strongly disagrees and holds it is De'oraysa even now, but he does not address the question of Avoda. Similarly, the Keren Ora there in Sota (D'H Ve'od) raises the question of avoda, and says that he would like to think that Birkas Kohanim is De'Rabanan now for lack of Avoda, but admits that it is clear that the Rambam and others hold it is De'Oraysa, as his brother, Reb Yaakov Karliner says in Teshuvos Mishkenos Yaakov OC 84. So what do you do with Reb Yaakov Emden and the Keren Orah's question?
With this we understand the nusach we use to introduce Birkas Kohanim on Yomtov- ותערב עליך עתירתינו כעולה וכקרבן. The Gemara in Sotah says that the Kohanim must begin moving toward the platform during Re'tzei, because Re'tzei is the Bracha of Avodah, and Duchenning must follow Avodah. (We delay the bracha until in middle of Hoda'ah, the Gemara says, because Avodah and Hoda'ah are essentially twins.) The reason we says Ve'Seiareiv is to emphasize this connection: We are asking that Hashem accept our tefilla as if it were a real Korban. Now that we have brought our Korban Tzibur, we are allowed to proceed to the Duchenning.
I have to mention, though, that the Mishna Berura very strongly asserts that Duchening is d'oraysa bizman hazeh. See the Shaar Tziun in 128, and also see Igros Moshe OC 5:20:23.
So the question remains, and this is where we were headed the whole time:
It seems that there is a machlokes here whether the rule of "tefilla=korbanos" satisfies Birkas Kohanim's prerequisite of avoda. What is the basis of the machlokes? What is the yesod of the machlokes? I think that to answer this question we need to know WHY the prerequisite exists, and what it means, what it says about Birkas Kohanim and what it says about the Kohanim who are saying Birkas Kohanim. And let's not forget the Mishneh Le'Melech.
TBC, maybe. You really have everything you need to work out your own mehalach.