Try using 2 cups of shredded, roasted cauliflower to replace a couple of the eggs. (Roast the shredded cauliflower in the pan before adding the egg/cheese batter.)

You can also try adding additional Italian spices / chopped green onion / oregano / minced garlic, etc, to overpower the eggy taste, if that is what you find offensive.

You could experiment with tofu, soy protein isolate, xanthan gum, and / or flaxseed meal, but I have no idea how that might turn out.