While the pasta for this dish is cooking, heat the olive oil and butter in a large skillet over moderate heat. Add the garlic and cook, stirring, for 2 minutes. Add the eggs and cook, sunny-side up, basting with a spoon for 2 to 3 minutes without turning, or until the eggs are just done. Remove the pan from the heat and sprinkle eggs with coarse salt and hot red pepper flakes.
Time the dish so that the pasta is ready as soon as the eggs are done. Place the pasta on a serving platter and toss with the Parmagiano-Reggiano, additional pepper flakes and parsley. Pour the contents of the egg pan, including the oil, butter and garlic onto the pasta. Garnish pasta and eggs with oregano or sage and serve immediately.