Jeffersons Vegetable Layer Cake

Name: Jefferson's Vegetable Layer Cake
Ingredients: 3 red bell peppers, halved lengthwise, stemmed, seeded and flattened; 2 medium zucchini, trimmed and sliced lengthwise 1/4 inch thick; 1 medium yellow squash, trimmed and sliced lengthwise 1/4 inch thick; 1 medium eggplant, trimmed, peeled, and sliced 1/4 inch thick; 1/4 cup olive oil, plus more for serving; Coarse salt and freshly ground pepper; 2 red tomatoes, sliced 1/4 inch thick; 20 medium fresh basil leaves, preferably Thai basil; 2 mangoes, peeled and sliced 1/4 inch thick; 2 yellow tomatoes, sliced 1/4 inch thick; 5 sheets gelatin; 1 cup Tomato Water; 5 ounces fresh goat cheese; Aged balsamic vinegar, for serving