In a 12-inch frying pan with sides at least 2 inches tall, or a 5- to 6-quart pan, heat 1 1/2 tablespoons olive oil over medium-high heat. Add onion and garlic and stir frequently until onion is limp, about 5 minutes.
Add tomatoes (with juice), eggplant, 1/2 teaspoon salt, 1/4 teaspoon pepper, and 1/2 cup water and bring to a simmer. Cover, reduce heat, and simmer, stirring occasionally, until eggplant is soft when pierced, about 10 minutes.
Stir in bell peppers, zucchini, and yellow squash. Return to a simmer, cover, and cook until squash is tender when pierced, 8 to 10 minutes longer.
Stir in basil and remaining 1 1/2 tablespoons olive oil. Add more salt and pepper to taste.