Pour tomato purée into a 10-inch long oval baking dish. Drop in the onion and garlic, stir in 1 tablespoon of the olive oil, and season the sauce generously with salt and pepper.
Trim the ends off the eggplant, zucchini, and squash. Carefully trim the ends off the red pepper and remove the core, leaving the edges intact.
On a mandoline, or with a very sharp knife, cut the eggplant, zucchini, squash, and red pepper into very thin slices.
Arrange the sliced vegetables on top of the tomato purée concentrically from the outer edge to the inside of the baking dish. Alternate the vegetables, and overlap them so just a tiny part of each flat surface is visible.
Drizzle the remaining tablespoon of olive oil over the vegetables, and season them generously with salt and pepper. Remove the leaves from the thyme sprigs, and sprinkle them over the dish.
Cover the dish with a piece of parchment paper cut to fit inside the dish.
Bake for approximately 45-55 minutes, until the vegetables have released their liquid and are cooked, but still hold their structure. They should not be brown around the edges, and the tomato sauce should be bubbling up around them.