Heat the oil in a large saute pan over medium-high heat. Add the onions and saute, stirring occasionally, until softened, 5 to 7 minutes. Reduce the heat to medium low and add the garlic, fresh and canned tomatoes, green and red peppers, piment d'espelette, sugar, 1 teaspoon salt and the balsamic. Cook, stirring occasionally, until the tomatoes have broken down and the peppers are very soft, 15 to 20 minutes.
Pour the eggs into the pan and slowly stir until the eggs are cooked but not lumpy, 3 to 5 minutes. Remove from the heat and sprinkle with the parsley. Serve with crusty bread.