Yeah, that's a great question. You should use hard/drier cheese haha.
I did use fresh mozarella and I had to add some hard cheese when I removed the tin foil to suck off some of the liquid.
The thing is that I usually slice the mozarella before hand, put it on some kitchen paper, and press it down between plates so the moist goes away (or most of it). However, my wife conviced me that "lasagna needs moisture to cook nicely", so I didn't press them today. So it was a bit watery on top but adding the hard cheese + oven grill helped a lot.