Heat the oven to 350°F and butter a 9-by-13-inch dish. Spread the biscuit quarters in the dish and bake for 8 minutes.
Brown the sausage in a skillet. Sprinkle in the flour and cook for 1 minute, then stir in 2 cups milk gradually. Add 1/2 teaspoon each salt and pepper and simmer until lightly thickened.
Whisk the eggs with the remaining 1/2 cup milk, salt and pepper. Pour around the partly baked biscuits.
Spoon the sausage gravy evenly over the eggs and sprinkle with cheddar.
Bake for 35 to 40 minutes more, until the center reaches 160°F and the biscuit tops are browned. Rest 10 minutes before serving.