What should've been played after this?

1 e4 d5 2 exd5 Nf6 3 d4 Bg4 4 Nf3 Qxd5 5 Be2 e6 6 a3?! Nc6 7 Nc3 Qf5 8 O-O Bd6 9 Be3 Ne4

One of the nice things about this opening for white is he can play c4 (the c3 knight isn't blocking the pawn). So 7.Nc3 definitely feel out of place.

Generously, black spends a whole tempo to put his knight on e4 and lets white exchange there and then play c4 afterwards. Black will lose another tempo when the e4 queen has to move again at some point in the future. Basically black loses two tempi in one move

Engine agrees with Nxe4 + c4.

On 10.Bd3 black can capture on f3 and if white recaptures with the pawn black has multiple ways to avoid the pin 10.Bd3 Bxf3 11.gxf3

11...Qg6+ or 11...Qh3 or 11...Nxc3 (this one wins a piece for black).