Considering 1.e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nxf7 Kxf7 5. d4 Be6. Firstly: as a white player you still have nine other moves than 5.- Be6 to consider, especially (in my opinion) my favourite 5.- c5. So how would you meet them?

Anyway - after 5.- Be6 Yusupov in his heavyweight Petroff "bible" discusses both 6. Nc3 and 6. Bd3. He quotes Staunton (!) for the line 6. Nc3 Be7. 7. f4 "with attack" (f4 is a standard white move). Concerning 6. Nc3 he mentions a game Shirazi-Lane. I have found the game in ChessAssistant and it is a very type Cochrane white success game:

[Event "Ch USA"]
[Site "Los Angeles (USA)"]
[Date "1983.??.??"]
[Round "?"]
[White "Shirazi Kamran (USA)"]
[Black "Lane Gary W (AUS)"]
[Result "1-0"]
[ECO "C42"]
[WhiteElo "2470"]
[BlackElo "2423"]
[Annotator ""]
[Source ""]
[Remark ""]

1.e4 e5 2.Nf3 Nf6 3.Nxe5 d6 4.Nxf7 Kxf7 5.d4 Be6 6.Bd3 Be7 7.O-O g6 8.c4 Kg7 9.Nc3 Na6 10.a3 c6 11.h3 Nc7 12.d5 Bf7 13.Be3 h6 14.f4 c5 15.e5 Ng8 16.Ne4 Bxd5 17.cxd5 Nxd5 18.Bd2 dxe5 19.fxe5 Qd7 20.Bc4 1-0

I have walked quickly through the game with a chess engine: white does not have much around move 10, but later the wellknown chess author mr. Lane goes terribly wrong, 15.- c5 allows the e4-e5 breakthough, 15.- Nd7 was much better, and 17.- Nxd5 is harakiri. But the game shows very typical way of black going down in this gambit even from a quite good position.

I do not care very much myself about a scientific/theoretical discussion on whether white can get an advantage against the Petroff. That might be important at e.g. 2400 level and above, but not for me (ELO 1993). My idea is to play basically sound openings (i.e. not e.g. the CG) and by preparation hopefully know them better than my opponent. 

But concerning the question on advantage against the Petroff, there is the new very fine Beating the Petroff by Kotronias and Tzermiadianos (5/5 review by Carsten Hansen at ChessCafe). This is very narrow white repertoire book, narrow in the sense that is seldom gives white more than one move. The book recommends a main line approach 3. Nxe5 d6. 4. Nf3 Nxe4. 5. d4 d5. 6. Bd3 Be7. 7. 0-0 Nc6. 8. c4 Nb4. 9. Be2 Be6. 10. Nc3 0-0. 11. Be3 Bf5. 12. Rc1 and black deviations from this. 

The book has a 27 pages introduction Typical ideas and Manoeuvres, which is very well written and interesting for black players as well. So if you are prepared to play positions like these, this book may give you a weapon - and maybe the answer to the "advantage discussion". 

But on the other hand: The line featured in the book probably represents exactly the kind of positions which make a lot of people feel sick of the Petroff. Since you are a BDG player the line may not be of your taste.

One the highly praised Khalifman Opening Repertoire According to Anand books also discusses how to play white against the Petroff. I don't have the book, but a good guess is that it also deals with some main line approach.

Perhaps you should consider 3. Bc4, which can lead to dynamic positions. There are also lines such as:

3. d4 Nxe4. 4. Bd3 d5. 5. Nxe5 Nd7. 6. Nxd7 Bxd7. 7. 0-0 Qh4. 8. c4 0-0-0. 9. c5 g5 with the board soon getting on fire with chances for both sides. Here you have to know some black deviations as well, but there are not as may as you may think.

The line 3. d4 Nxe4. 4. Bd3 d5. 5. Nxe5 Nd7. 6. Qe2 Nxe5. 7. Bxe4 dxe4. 8. Qxe4 Be6. 9. Qxe5 is also quite sharp, but it is black who is the gambit player getting initiative for the pawn.

A line often recommended for white players who want to get away from the main road is 3. Nxe5 d6. 4. Nf3 Nxe4. 5. Nc3, which is analyzed in one of the wellknown white repertoire books (sorry I don't remember which).

Have you considered the KG - it is much more than the CG a gambit of the same type as the BDG (and no words from me about soundness  )?

It is no easy matter to prove an advantage against any serious defence, and the Petroff is a particularly tough nut to crack, let's face it.  It seems to me that as a BDG player you are better off looking for a line which leads to reasonably open positions and avoiding the somewhat stodgy main lines.

Obviously the Cochrane is one way to go, and if you are addicted to sacrificing material then fine, go for it: it should yield you at least some interesting games, and it is not a trivial matter for Black to organise his forces.  On the other hand you are certianly giving Black more winning chances than he is used to in the Petroff ...

Another way to go, though, which does not involve what is, after all, significant material investment, is the line used a bit by Shirov recently and recommended for White in Sam Collins' recent 1.e4 repertoire book, namely 3.Nxe5 d6 4.Nf3 Nxe4 5.Nc3!? ^ 5...Nxc3 6.dxc3 and then e.g. 6...Be7 7.Bf4 0-0 8.Qd2 Nc6 9.0-0-0!? with opposite side castling and free development.  I doubt that it is a serious theoretical threat to Black, but at club level it looks very serviceable and avoids a hell of a lot of heavy Petroff theory, without giving up a knight!

On the Cochrane Gambit (CG) in general it is with a lot of other gambits. At grand master level it is surely unsound, but at club player level it may very well work a lot of times. (And I am not going to say anything about the BDG in that context  8)).

The CG is basically different from the BDG. In the BDG, white gives a pawn for better development, but with no initial weeknesses of the black position. In the CG, white gives a knight for two pawns, without having a development advantage, but instead a weekened black king and (in some lines) more space.

Another thing is that when playing the BDG, at club level at least, you are very likely to meet opponents that know very little of it. On the other hand with the CG, most Petroff players have faced it several times - that's my own experience as a Petroff player myself (did I really dare to admit that in this forum??  ), and any serious Petroff player should be prepared for it.

Then to the lines: Firstly, I am not able to guess what you mean by "1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nxf7 Kxf7 5. d4 Bc4!??!". It is surely not 5.- Bg4, because then you would most likely not think of playing 6. Bd3  .

Anyway: after 1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nxf7 Kxf7 white has three options: d4, Lc4+ and Nc3.

5. d4 is generally considered the best, but preparing this with white should take into account that black has 10 (ten!) relevant answers to choose from: Bg4, d5, Be6, Be7, Nbd7, Qe7, Qe8, c6, c5 and g6 - good luck with the preparation! Personally I prefer 5.- c5, which is very well analyzed in Survive and Beat Annoying Chess Openings by Schiller & Watson, and the lines would surely keep me away from playing the CG. Some sample lines:
6. Bc4+ d5. 7. exd5 b5! 8. Bxb5 Qxd5 9. 0-0 Bb7.
6. c3 cxd4. 7. cxd4 d5. 8. e5 Ne4. 9. Bd3 Bb4+.
6. Nc3 Nc6. 7. Bc4+ d5. 8. Nxd5 Na5! 9. Nxf6+ Nxc4. 10. Nd5 Be6.
6. dxc5 Nc6! 7. Bc4 Be6. 8. Bxe6+ Ke6 9. 0-0 d5. 10. e5 Nxe5. 11. Bf4 Bxc5. 12. Bxe5 Kxe5 ("Watch my here!", says the king). 13. Nc3 Kd6.

5. Bc4+ has a bad reputation due to 5.- d5, and e.g. 6. exd5 Bd6. 7. 0-0 Kf8. 8. d4 Kg8 or 6. Bb3 Bg4! 7. f3 Be6. 8. e5 Nfd7.

5. Nc3 is a new idea, which has been mentioned by MNb in the thread Winning against the Petroff!?. Black should probably answer with 5.- Be6. 6. d4 d5, e.g. 7. e5 Ne8. 8. Bd3 g6. 9. Qf3+ Kg7. The 5. Nc3 line may very well turn up to be the best idea for white, but will a BDG player really like the white position after 9.- Kg7?