Preheat oven to 375 degrees. Place the peppers, zucchini, squash, and eggplant, on rimmed baking sheets in a single layer. Drizzle with half of the oil, and season with salt and pepper. Transfer to oven, and bake for 10 minutes. Turn vegetables over, drizzle with remaining 2 tablespoons oil, and season with salt and pepper. Bake until tender and cooked through, 10 to 20 minutes more.
Layer the vegetables in a 9-inch springform pan, in the following order: eggplant, red pepper, yellow squash, red tomatoes, 10 basil leaves, zucchini, mango, and yellow tomatoes. Season with salt and pepper. Arrange remaining 10 basil leaves decoratively on top of the cake.
Soak the gelatin sheets in enough cold water to cover. In a small saucepan, place the tomato water over medium heat until hot to the touch. Squeeze the excess water from the gelatin sheets, and add to the tomato water. Stir until dissolved. Let cool slightly. Measure 1 cup; set remaining tomato water aside for another use. Pour evenly over the cake.
Wrap with plastic wrap, and transfer to refrigerator until set, about 1 hour.
Slice cake, and serve garnished with a leaf of basil, and a tablespoon-sized scoop of goat cheese. Drizzle plate with balsamic vinegar and olive oil. Serve immediately.