Preheat oven to 400°F. Line a baking sheet with aluminum foil; coat with cooking spray.
Place zucchini between a double layer of paper towels; squeeze/press to extract excess moisture. Mix together zucchini, turkey, onion, oregano, salt, pepper, basil, and egg in a bowl. Form 24 meatballs with turkey mixture; arrange on prepared baking sheet.
Bake at 400°F until fully cooked, about 12 minutes.
Meanwhile, heat a large ovenproof skillet over medium-high. Add oil to pan; cook mushrooms and garlic about 5 minutes. Stir in pizza sauce and 1/4 cup water; reduce heat to medium, and simmer 5 minutes. Lay meatballs in the skillet (leave behind cooked-out proteins); gently stir to coat with sauce. Add mozzarella cheese; gently stir.
Sprinkle with Sharp white cheddar cheese. Broil meatball mixture until cheese is melted and bubbly, about 2 minutes. Garnish with fresh basil; serve with toasted baguette.