Mist the eggplant rounds with cooking spray, then season both sides of each slice with salt and pepper. Arrange the eggplant slices in a single layer on the prepared baking sheet. Bake for 30 minutes, or until soft and golden brown. Remove from the oven and set aside.
While the eggplant cooks, in a large saucepan over medium, heat the olive oil. Add the onions and saute until softened and translucent, about 5 minutes. Stir in 1 teaspoon salt and the thyme, then reduce heat to low. Cook, stirring occasionally so the onions don’t burn, until very soft and browned, about another 30 minutes. Stir in the vinegar and remove from heat.
Mist an 11-inch springform tart pan (or a tart pan with a removable bottom) with cooking spray.
On a clean, floured surface using a floured rolling pin, roll the chilled dough into a 13-inch circle. Transfer the dough to the tart pan and fold in and press together the overhanging dough to build up the edges. If the dough tears or breaks, simply piece it together and press it into the pan.
Spread the onion mixture in an even layer over the bottom of the tart. Add an even layer of the eggplant. Top with tomato slices arranged in an overlapping circular pattern. Spray the top of the tart with olive oil cooking spray, then season with salt and pepper.
Bake until the crust is golden and the tomatoes are slightly browned, about 45 minutes. Remove from oven and cool on a rack. Remove the outer ring of the pan and transfer the tart to a serving plate. Slice into 8 wedges and serve warm, at room temperature or chilled. Serve garnished with torn basil leaves.
Nutrition information per serving: 340 calories; 160 calories from fat (47 percent of total calories); 18 g fat (2.5 g saturated, 0 g trans fats); 0 mg cholesterol; 41 g carbohydrate; 7 g fiber; 8 g sugar; 6 g protein; 610 mg sodium.