How to Make It
Heat the 3 tablespoons of olive oil in a large saucepan. Add the onion and cook over moderately low heat until softened and golden, about 15 minutes. Add the tomatoes and cook over moderate heat, stirring frequently, until the sauce thickens, about 35 minutes; if the sauce begins to stick to the pan, stir in some of the reserved tomato liquid. Season the sauce with salt and pepper; if it is tangy, stir in the sugar. Add the crushed red pepper and the basil and remove from the heat.
Preheat the oven to 200°. You will need 12 pieces of the pane carasau, either the packaged triangles or 7-inch squares. Heat the stock in a medium skillet. Warm 4 dinner plates in the oven.
Using tongs, dip a piece of the pane carasau in the hot stock, turning it once, until it begins to soften. Transfer to a warm plate. Spread a layer of tomato sauce over the bread and sprinkle with 1 tablespoon of the grated cheese. Repeat this layering 2 more times. Assemble the remaining 3 stacks on the warmed plates in the same way and keep warm in the oven.
Heat a thin film of olive oil in a large nonstick skillet. Crack in as many of the eggs as will fit comfortably and fry, over easy or sunny-side up; repeat with the remaining eggs. Set 2 fried eggs on each stack and sprinkle with grated cheese. Serve the pane frattàu immediately, passing more cheese at the table.