Place the eggplant, red bell pepper and tomatoes on a large sheet pan. Place the yellow squash, zucchini and shallot on another sheet pan. Drizzle with olive oil and season with salt and black pepper. Toss or flip so that the vegetables are evenly coated with oil, salt and pepper. Roast the zucchini, squash and shallot for 13-15 minutes and the eggplant, bell pepper and tomatoes for 20-25 minutes. Let cool to room temperature. Reduce the oven temperature to 350 degrees.
Cut the red bell pepper into 1/2-inch cubes and cut the tomatoes in half top to bottom.
Sprinkle the bottom of the tart with Parmesan cheese.
Place the eggplant over the Parmesan cheese. Sprinkle with half the red bell pepper, 1/3 of the goat cheese, 1/3 of the sliced basil and half the garlic.
Top with zucchini, yellow squash and shallot. Sprinkle with 1/3 of the goat cheese, 1/3 of the sliced basil and the remaining garlic.
Top with remaining red bell pepper, tomatoes, remaining goat cheese and basil.
Whisk the cream (or half-and-half) and eggs together. Add 1/2 teaspoon salt and 1/4 teaspoon black pepper.
Bake for 50-60 minutes or until filling set and cheese is lightly browned.
Let stand 10 minutes. Cut into wedges and serve warm.