How to play vs 1.e4 e5 2.Nf3 Nc6 3.d4 Nxd4? Assuming that we want to recapture, is it better to take the d4 Knight or the e5 Pawn?
After 1.e4 e5 2.Nf3 Nc6 3.d4 Nxd4, both 4.Nxd4 and 4.Nxe5 seem to lead to an advantage for white. However, it's hard to tell which of the two options is best. Therefore it's a matter of taste.
4.Nxd4 exd4 5.Qxd4 transposes to the Scottish main line, where black played the inferior 4....Nxd4. This option is the easiest for white: white stays in his repertoire after 3....exd4, unless he wanted to play a Scottish gambit line, e.g. 4.Bc4 or 4.c3.
However, 4.Nxe5 also yields an advantage for white. After 4....Nc6 5.Nxc6, white has an extra tempo compared to 3....exd4 4.Nxd4 Nf6 5.Nxc6. Instead, black can retreat with 4....Ne6. According to the Game Database of ChessTempo, 5.Bc4 is the most popular reply, but 5.g3 seems an interesting alternative.
Edit: It's important to mention that, after 4.Nxe5, 4....Bc5 is a mistake. One refutation is 5.Nxf7, given by Evargalo, as white regains the piece after 5....Kxf7 6.Qh5+. Instead, black should complicate matters by 5....Qe7 or 5....Qh4. According to Stockfish 9, play should continue with 6.Nxh8 Qxe4+ 7.Be3 Nxc2+ 8.Kd2. Another refutation is the natural 5.Be3. Once again the Nxf7-idea comes into play: 5....Ne6 6.Bxc5 Nxc5 7.Nxf7 Kxf7 8.Qh5+. Black can avoid material losses by 5....Qh4, as 6.Bxd4 is answered by 6....Qxe4+. Nevertheless, white obtains a clear advantage after 6.Nc3 Ne6 7.g3 Qf6 8.Nc4.
[StartPly "5"] [FEN ""] 1.e4 e5 2.Nf3 Nc6 3.d4 Nxd4 (3...exd4 4.Nxd4 (4.c3)(4.Bc4) Nxd4 (4...Nf6 5.Nxc6) 5.Qxd4) 4.Nxe5 (4.Nxd4 exd4 5.Qxd4) Bc5? (4...Nc6 5.Nxc6)(4...Ne6 5.Bc4 (5.g3)) 5.Be3 (5.Nxf7 Qe7 (5...Kxf7 6.Qh5+)(5...Qh4 6.Nxh8 Qxe4+) 6.Nxh8 Qxe4+ 7.Be3 Nxc2+ 8.Kd2 Bxe3+ 9.fxe3 Nxa1 10.Nc3) Qh4 (5...Ne6 6.Bxc5 Nxc5 7.Nxf7 Kxf7 8.Qh5+) 6.Nc3 (6.Bxd4 Qxe4+) Ne6 7.g3 Qf6 8.Nc4