In a skillet, cook sausage over medium heat until browned, breaking it into small pieces. Remove from heat and let cool slightly.
Whisk 5 eggs, milk, salt, and pepper in a bowl until well combined.
Mix the cooked sausage, egg mixture, and shredded cheese in a large bowl.
Preheat the oven to 350°F (175°C). Grease a 9×13-inch baking dish and evenly pour in the sausage mixture.
Arrange buns around the edges of the dish. Bake for 20-25 minutes.
Remove from oven, crack the remaining 5 eggs over the top and bake for an additional 5 minutes or until eggs are set. Garnish with parsley and serve warm.