Kaufman recommends the Giuoco Piano rather than the Two Knights gambit. His lines run deeper than the three tabiya I will present here, as these positions are for searching for master games from TWIC weekly set. Giuoco Piano is Italian for quiet game. It is quieter than the Two Knights, but I think it a bit of a misnomer.
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 is the Giuoco Piano. Like most open games, black will try to preserve a pawn on e5 and inhibit white from playing d4. Black gets his dark square bishop out of the coming pawn chain immediately. Also, it allows black to castle short immediately after playing ...Nf6 to protect f7.
First is the Evan's Gambit. A solid gambit, but black should be able to equalize. The usual idea of taking the gambit and returning the pawn at the right moment works, but Kaufman recommends a line where black tries to hold onto the pawn. 4.b4 Bxb4 5.c3 Ba5 6.d4 d6 7.Qb3 Qd7 8.O-O Bb6
Black will support the e5 pawn with f7-f6, develop his Ng8-e7, and castle long after unwinding his queenside. The awkward placement of the Qd7 is necessary to guard f7 and to be able to retake on c6 with the queen rather than the pawn if white plays Bb5 and Bxc6. Unwinding ideas are Na5 to take the Bc4, Qc6, and Be6 or Bg4.
The mainline is 4.c3 to support a d4 push, but White can transpose to the Italian Four Knights or castle and play a risky gambit. This is difused by 4.O-O Nf6 5.d4 Bxd4 6.Nxd4 Nxd4 7.f4 d6 8.fxe5 dxe5 9.Bg5 Qe7.
There are two approaches white can take after 4.c3 Nf6. He can strike at the center with 5.d4 or play the Giuoco Pianissimo 5.d3 O-O 6.O-O d6.
This is the line played most by masters these days. Like the d3 Ruy lines, white still plans to play d4 later after some preparatory moves. Unlike the Ruy, black has not made the weakening/space gaining pawn moves on the queenside. Black can make use of these moves to develop, which white will use to guide his plans, which center around the possibility of a kingside attack with pieces.
The old mainline is still played, but it is heavily analyzed.. 5.d4 exd4 6.cxd4 Bb4+ 7.Bd2 Bxd2+ 8.Nbxd2 d5 9.exd5 Nxd5.
We have an isolani game with all the normal imbalances.