Can White play e4 followed by f3? We know that 1. e4 e5 2. Nf3 f6 is a disaster for Black (Damiano Opening).

But White has an extra move. Can he therefore play 1. e4 e5 2. f3 ?

If Black replies 2... Nf6, White has 3. Ne2 (followed eventually by Ng3). That seems to stave off the immediate problem of the Damiano Opening.

So does the extra move give White enough leeway to play like this?
2.f3 looks like a terrible move. It wastes a tempo by not developing a piece; it takes f3 square away from the knight, and it weakens a7-g1 diagonal, while achieving absolutely nothing. I think that after any normal move (2...d5, 2...Nc6, 2...Nf6, 2...Bc5) black has the edge.

While I don't see any reason to give up the advantage so early, I don't think white is lost (just slightly worse), so white can play it if they want to.

I also think that 1. e4 e5 2. Nf3 f6 leads to a "disaster" is an overstatement. 1. e4 e5 2. Nf3 f6 3.Nxe5 fxe5?? does lead to a disaster; however, black can play 3...Qe7.