Cook the sausage in a large skillet over medium-high heat, breaking it up, until browned and cooked through, about 8 minutes. Transfer to a sheet pan to cool.
In the same skillet, cook the hash brown potatoes according to package directions until golden and crisp, about 12 minutes. Add to the sheet pan with the sausage.
Whisk the eggs with the milk, salt, and pepper. Melt the butter in the skillet over medium-low heat and scramble the eggs slowly, pulling them off while still slightly soft and glossy. Spread on the sheet pan.
Let everything cool to room temperature, about 20 minutes. Do not skip this — warm filling makes soggy frozen burritos.
Warm the tortillas briefly so they roll without cracking. Divide the eggs, sausage, potatoes, and cheese among them, keeping the filling in a line slightly below center.
Fold in the sides, roll tightly, then wrap each burrito in foil. Place the wrapped burritos in a large zip-top freezer bag, label with the date, and freeze.