The ratatouille filling can also be served as a side dish for scrambled, fried, or poached eggs. If you like, you can cook the vegetables in separate pans at the same time.
Heat 1 tablespoon olive oil in a 12-inch nonstick saute pan over medium heat. Add onion, and season with salt and pepper. Cook, stirring frequently, until onion is soft and translucent, about 8 minutes. Add tomatoes, season again with salt and pepper, and cook slowly until the tomatoes stop releasing juices, about 5 minutes. Transfer mixture to a large bowl; wipe pan clean.
Return pan to medium heat. Add 1 tablespoon olive oil. Add squash, and season with salt and pepper. Cook until squash is brown but still firm, about 4 minutes. Transfer mixture to bowl with tomatoes; wipe pan clean.
Return pan to medium heat. Add the remaining 2 tablespoons olive oil. Add the eggplant, and cook until it is browned on all sides and completely soft, about 3 minutes. Return tomato mixture to pan. Simmer over low heat, stirring occasionally, about 10 minutes. Stir in parsley and basil, and season with salt and pepper. Return mixture to bowl, and set aside. Wipe pan clean.
Whisk together 6 eggs and 1 tablespoon water in a medium bowl. Season with salt and pepper. Melt 1 tablespoon butter in the saute pan over medium heat. Add the egg mixture; as eggs begin to set, push them toward the center of the pan with a rubber spatula, letting the uncooked part run underneath to fill the void.
When eggs are cooked but not completely dry, slide omelet onto a baking sheet. Spoon half the filling along one long edge; with your hands, carefully roll into a neat log. Using a large metal spatula, transfer omelet to a serving platter with the seam side down. Keep warm.
Repeat process with the remaining ingredients. Cut each log on the diagonal into individual pieces, and serve.