Preheat the oven to 375 degrees F.
Pour the tomato puree into a 10" baking dish. Drop in the sliced garlic and chopped onion, stir in one tablespoon of the olive oil and season generously with salt and pepper.
Trim the ends off the eggplant, zucchini and yellow squash. Slice them, and the tomatoes, into very thin slices (about 1/8" thick). Carefully arrange the slices of prepared vegetables on top of the sauce in neat rows, overlapping them so that just a bit of each flat surface is visible, alternating vegetables. (It's okay if you have leftovers that don't fit.)
Drizzle the remaining olive oil over the vegetables and season them generously with fresh thyme, salt and pepper. Cover the dish with a piece of parchment paper cut to fit inside.
Bake for approximately 45 to 55 minutes, or until veggies have released their liquid and are clearly cooked, but with some structure left so they are not totally limp. They should not be brown at the edges, and the tomato sauce should be bubbling. Serve with some crusty French bread, or on top of polenta or couscous.
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