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 (the yolks must remain runny). Remove the pan from the heat and sprinkle eggs with coarse salt and hot red pepper flakes.
You must time this 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 Parmigiano-Reggiano, additional red 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. Pass the additional Parmigiano-Reggiano at the table.
Suggested drink: Giuseppe Franca Nozze D'Oro, Tasca D'Almerita, 1994.