In a small bowl, combine chicken, squash, egg, lemon zest and 1/4 cup of the bread crumbs. Season with salt and pepper and mix well.
Heat the oil in a skillet. Meanwhile, form the chicken mixture into 8 patties about 2 inches in diameter and an inch tall.
Pour remaining breadcrumbs in a plate and coat patties in crumbs. Fry in the oil until golden brown on each side, flipping once. Drain on paper towels.
Turn on the broiler. Top each patty with a half slice of cheese and place under broiler until cheese has melted.
For the sauce, heat the oil over medium high heat in a skillet. Saute onions until softened, about 5 minutes.
Add in mushrooms and garlic and saute until mushrooms have softened, another 5 minutes.
Toss the tomatoes into the skillet just long enough to heat up. Season with some salt and pepper. Transfer to a bowl and stir in the basil.
Serve each person 2 patties and top with the tomato and mushroom sauce and some grated Parmesan cheese.