In a large, heavy-bottom sauté pan, heat ¼ cup of the oil over high heat. Add the onion and garlic; when the onion is translucent and just beginning to brown, reduce the heat and stir in the tomato paste and thyme.
Remove from the heat and transfer the mixture to a sheet pan to cool.
Wipe out the sauté pan (or use a clean one) and heat two tablespoons of the oil in it over high heat. Add the zucchini and sauté over high heat until tender and beginning to become golden brown at the edges. Season with salt and pepper, remove from the heat, and transfer to the sheet pan with the onion mixture, keeping it segregated.
Repeat this procedure with another one to two tablespoons of the oil as needed, and the squash, pepper, and eggplant, cooking each vegetable separately, seasoning with salt and pepper, and allowing it to cool on one of the sheet pans. Wipe out the pan between vegetables.
Once the vegetables have cooled to room temperature, combine them in a large mixing bowl and toss gently. Add the balsamic and remaining oil to taste. Adjust the seasoning with salt and pepper, add the basil, and let the mixture sit at room temperature for three to four hours to let the flavors marry. Serve at room temperature.