Preheat oven to 425°F. Grease a 9-inch deep-dish pie plate and place it on a baking sheet.
Squeeze thawed hash browns until very dry. Mix with butter, seasonings, parmesan (optional), and egg white (optional).
Press potatoes firmly into the pie plate, across the bottom and up the sides.
Bake crust 20–25 minutes until golden at the edges and set on the bottom. Reduce oven to 350°F.
Cook turkey sausage in a skillet until browned; drain on paper towels.
Whisk eggs, half-and-half, Dijon (optional), salt, pepper, and paprika (optional).
Sprinkle half the cheddar in the crust. Add sausage and green onions. Top with remaining cheddar.
Pour egg mixture into the crust.
Bake 40–50 minutes until edges are set and the center gently wobbles.
Rest 10–15 minutes before slicing.