Line two baking sheets with parchment paper. Preheat oven to 400 F. Blend olive oil with thyme, salt and pepper. Set aside. Prepare veggies according to the directions below. Cut all of the veggies into slices between 1/4 and 1/2-inch thick.
Eggplant - lightly salt both sides of the slices and lay them on a rack to allow them to sweat for about 15 minutes. Pat them dry on both sides with a paper towel.Onions - peel and slice Bell peppers - remove stem and seeds from the bell pepper and cut into pieces that are about 2-3 inches wide. Tomatoes - remove stem and slice.Zucchini – slice into pieces about 3-inch long. You will need two slices per stack. Portobello caps – clean caps and remove stem. Lay all of the sliced veggies onto the two prepared baking sheets and lightly brush the tops with the olive oil mixture.
Roast veggies until al dente, about 20 minutes. Remove from oven and cool. Once the veggies are cool enough to handle, line a baking sheet with a new piece of parchment paper and layer each stack on the parchment, beginning with eggplant on the bottom, then onions, tomatoes, mushrooms, bell pepper, goat cheese, and zucchini. Secure each in place with a toothpick through a basil leaf.
Bake before serving for about 20 minutes at 400 F or until veggies just begin to brown and cheese is soft and melty. If you want the basil leaf to look pretty when you serve it, put it on after baking the stacks.
*Note – you can place the veggies in any order you like, but I like to use the most firm ones on the bottom.