Bundle the thyme and basil together with butcher's twine, then set aside. Heat half the olive oil in a large skillet over medium heat. Add the eggplant and begin to cook, browning lightly on each side, 4 to 5 minutes total. Add about a quarter of the garlic and sprinkle with salt and pepper. Transfer the eggplant to a cold Dutch oven. Return the skillet back to the heat, then add more olive oil and the onions and peppers. Cook, browning lightly on each side, 4 to 5 minutes total. Add about a quarter of the garlic and sprinkle with salt and pepper. Transfer to the cold Dutch oven with the eggplant. Finally, repeat this process with the zucchini, then add to the Dutch oven. Stir the anchovy paste, tomatoes, olives, remaining garlic, herb bundle and salt and pepper into the vegetables. Drizzle with olive oil. Stir, then cover and bake 1 hour.
Remove from the oven. Transfer to a platter and garnish with Parmesan and basil. Serve by itself as a side dish, or serve over pasta, eggs or polenta.