For the Pangritata

1. In a large fry pan, heat 2 tablespoons of olive oil over medium-high heat.
2. Add the breadcrumbs to the pan and sauté until golden and crispy, about 4 to 5 minutes.
3. Stir in the chopped rosemary, then remove from heat quickly to prevent burning.
4. Once cooled, mix in the lemon juice and set aside.

For the Pasta

1. Bring a large pot of well-salted water to a boil and cook the spaghetti until al dente, according to package instructions.
2. Wipe out the skillet used for pangritata, add 1 tablespoon each of olive oil and butter, and melt over medium heat. Add garlic, then carefully break the eggs into the skillet. Adjust the heat to avoid burning the garlic while allowing the egg whites to set but keeping the yolks runny.
3. Drain the pasta, reserving ½ cup of the cooking water. Return the pasta to the pot, then add the garlic, eggs, and fat from the skillet, parsley, and capers. Toss well, breaking up the eggs. Use the reserved pasta water to help combine the ingredients smoothly.
4. Plate the pasta, season with black pepper and salt, sprinkle over with grated cheese, and top with the prepared pangritata.