Spaghetti with Caramelized Onions and Fried Eggs

Heat the oil in a large skillet over medium-low heat. Add the onions, reduce the heat to low and cook, stirring occasionally, until golden, about 15 minutes. Stir in the garlic, red pepper flakes, salt to taste and a grinding of pepper. Set aside. Cook the spaghetti in a large pot of boiling salted water until al dente, 10 to 12 minutes. If using the broccoli, add it during the last 4 minutes of cooking. Meanwhile, reheat the onions in the skillet over medium-low heat. Break the eggs on top of the onions, spacing them evenly, and cook until the whites are set but the yolks are still runny, about 5 minutes. Sprinkle with salt and pepper. Drain the spaghetti and broccoli, if using, and immediately return to the cooking pot. Add the egg and onion mixture and the cheese and stir thoroughly until the yolks coat the spaghetti and the whites break up into pieces. Serve at once, with additional cheese on the side.