Heat the oven to 375°F and grease a 9-by-13-inch dish. Brown eggplant in 1 tablespoon oil and transfer to a bowl; repeat with zucchini and another tablespoon oil.
Add remaining oil, peppers and onion and cook 7 minutes. Add beef and brown, then drain. Add garlic and tomato paste for 1 minute.
Stir in tomatoes, thyme, basil, salt and pepper, return eggplant and zucchini and simmer 5 minutes.
Spread in the dish and top with mozzarella and Parmesan.
Bake for 30 to 35 minutes until bubbling and browned. Rest 10 minutes.