You must time this dish so that the pasta is ready as soon as the eggs are done.
Cook pasta according to package directions to al dente. Learn How To Cook Pasta Properly.
While the pasta for this dish is cooking, heat the olive oil and butter in a large frying pan over medium 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 should still be runny and the whites, while set, still soft). Do not overcook the eggs.
Remove the pan from the heat and remove the eggs to a warmed plate; sprinkle eggs with coarse salt and hot red pepper flakes. Learn how to make perfect Fried Eggs.
When the pasta is done, drain and toss with 1 tablespoon olive oil, red pepper flakes, and parsley.
Divide the pasta among individual warmed serving bowls or plates. Pour the contents of the egg pan (including the oil, butter, and garlic) onto the top of each individual pasta. Sprinkle with the grated Parmesan cheese. Top each portion with a fried egg.
If desired, garnish pasta and eggs with minced oregano or sage.
Serve immediately with additional Parmesan cheese at the table.
NOTE: Each diner should break up the egg and mix it into the pasta to create a sauce.
Makes 4 servings.
Linguine with Fried Egg and Herb Topping Recipe: https://whatscookingamerica.net/pasta_rice_main/linguinefriedegg.htm