Combine 1 cup broth, grated cheese, and next 3 ingredients in a medium bowl; stir with a whisk until eggs are lightly beaten.
Bring remaining broth to a boil in a Dutch oven over medium-high heat. Reduce heat to low; gradually pour egg mixture into broth, stirring constantly with whisk (the faster you stir, the finer the pieces of egg). Cook 3 minutes, stirring constantly.
Remove from heat; stir in pepper, salt, and nutmeg.
