Gently pound cutlets between 2 sheets of plastic wrap with a flat meat pounder or a rolling pin until 1/4-inch thick.
Whisk together eggs, water, salt, pepper, cheese, and parsley in a shallow bowl (batter will be thick). Put the flour on a plate.
Heat 1 tablespoon butter and 1 tablespoon oil in a 12-inch nonstick skillet over moderately high heat until foam subsides. Working quickly, dredge 1 cutlet in flour, then in the batter, letting excess drip off, and add to skillet. Repeat with about 3 more cutlets (do not crowd in skillet) and cook, turning over once, until golden and just cooked through, about 4 minutes total. Repeat with remaining butter, oil, and cutlets. Serve with lemon wedges.