Italian game - Nxc3
Try playing it against the engine and see how it refutes every single try that you can come up with as Black
You can be more direct there. Ng5 to Qf3 and this is all over for black. Qxc3 is unstoppable because black is forced to protect f7, and the bish can't do that.
You can even start dreaming about 2 substitute sacs - a rook on e1 to get a rook on e8, and the rook on e8 to give a queen mate on e8 with Bxh7-but that is a pipedream at this stage
That's why I play 1. d4 - this stuff is just too easy.
I see what you are talking about. The move you are describing is not a part of the Moller Attack, instead, it is called the Greco variation of the Greco Gambit. Nxc3 isn't a mistake, but Black should tread carefully in that position. If Black takes the pawn, then White is winning. You should do some of your own analysis or research in that position (Chessfactor and Remote Chess Academy's videos are good in this position (I'm too lazy to find the link and paste it here.)