Heat a large skillet over medium-high heat; cook and stir sausage, breaking meat into chunks, until crumbly and no longer pink, about 10 minutes. Transfer sausage to a paper towel-lined plate to drain.
Mix cooked sausage, mozzarella cheese, Parmesan cheese, and egg together in a bowl.
Rough dough into a 10x14-inch rectangle on a baking sheet. Spread sausage-cheese mixture onto dough, leaving a 1-inch border. Roll dough around filling, starting from the long edge, creating a long log. Form the log into a circle and pinch edges together to seal. Brush top of roll with egg yolk.
Bake in the preheated oven until ring is dark golden brown, 1 hour. Cool for 15 minutes before slicing into pieces to serve.