Preheat oven to 400°F. Line 2 large baking sheets with foil. Mist foil with cooking spray.
Warm oil in a large skillet over medium-high heat. Add onion and sauté until tender, 3 to 5 minutes. Add garlic and sauté 1 minute longer. Stir in tomatoes and tomato paste; cook, stirring often, until thick and pulpy, about 10 minutes. Add turkey and cook, stirring often and breaking up any large clumps with a spoon, until most of liquid has evaporated, 10 to 15 minutes. Stir in thyme and season with salt and pepper.
Using a serrated knife, cut each pita in half horizontally. Place rounds, cut side up, on baking sheets. Divide turkey mixture among rounds (about 1/2 cup each) and sprinkle with mozzarella and Parmesan, if desired. Bake until cheese has melted and pizzas are lightly browned, 12 to 15 minutes.