Toss cubed eggplant with a good sprinkle of salt, place in a strainer and let drain while you prepare the other ingredients.
Heat 1/4 cup olive oil in a large skillet over medium high heat. Add onions and garlic and cook until soft, about 6 minutes then rinse eggplant and add to skillet. Cook stirring often until eggplant has browned, about 8 minutes then add red pepper, green pepper, squash, bay leaves, and red pepper flakes and continue to cook until vegetables have begun to soften, about 6 minutes then add tomatoes and cook covered until all the vegetables have softened and the mixture has begun to thicken, about 12 minutes. Remove from heat and season to taste with salt and pepper.
Heat remaining oil in a non-stick skillet, and when shimmering add eggs and fry until whites are set and yolk is runny. If you’re making a batch of ratatouille to eat all week, store ratatouille in an airtight container, and then reheat and top with a fried egg for breakfast as desired.