The name of which cheese is Italian for twice-cooked as made not from curds but the whey other cheeses?

The cheese you're thinking of is Ricotta.

"Ricotta" is Italian for "recooked" and it's indeed made from the whey left over after other cheeses (like mozzarella or Parmesan) are made. It's not technically made from curds, but rather the proteins that remain in the whey.