Bring broth to a simmer in a pot over high heat. Taste broth and adjust salt and pepper as needed.

Whisk eggs, grated Parmigiano Reggiano and Pecorino Romano cheeses, parsley, semolina flour, salt, pepper, cayenne, and nutmeg together in a bowl until well blended.

While stirring the simmering broth, slowly pour egg mixture into the broth. Bring mixture back up to a simmer. The liquid will be slightly cloudy initially and clear as broth heats.

Serve in bowls; garnish with a drizzle of olive oil and a pinch of red pepper flakes.