1. Heat oven to 250 degrees. Cut tomatoes in half lengthwise and arrange cut side up on a parchment-lined baking sheet. Brush with olive oil and sprinkle liberally with salt, pepper, and thyme; sprinkle with sugar. Place in oven. If you have a gas oven with a pilot light, turn off heat after 2 hours and leave tomatoes in overnight. If you have an electric oven, bake for 2 hours, then turn to "warm" setting and leave tomatoes in overnight.

2. Cut eggplants in half lengthwise and trim tops. Salt liberally and place on a rack cut side down; let sit for 30 minutes. Rinse and pat dry.

3. In a 10-inch nonstick pan with an ovenproof handle, heat 1 tablespoon olive oil over medium heat and add onion and garlic. Cook until onion is soft, translucent, and golden brown, about 5 minutes. Remove from pan and set aside. Turn up heat to medium high. Add 2 teaspoons olive oil and sear eggplant, cut side down, until brown, about 5 minutes. Turn and cook until very tender, another 5 to 10 minutes. Remove eggplant and set aside.

4. Add 2 teaspoons olive oil and cook zucchini slices until brown on both sides, about 5 minutes. Season with salt and pepper. Cut in half diagonally and set aside.

5. Return all but 2 tablespoons of the onion-garlic mixture to pan. Layer eggplant and zucchini on top of onions. Arrange tomatoes over eggplant and zucchini. Season well with salt and pepper.

6. In a small bowl, whisk together eggs and milk or half-and-half; season with salt and pepper. Tear basil leaves into pieces and add to egg mixture. Pour mixture over vegetables. Crumble goat cheese into large pieces and scatter over frittata along with olives.

7. Heat oven to 350 degrees. Cook frittata on stove over low heat until set around the edges, about 10 minutes. Transfer to oven and bake until top is set, about 5 to 10 minutes. Garnish with reserved onion and garlic. Remove from pan and cut into wedges.