Heat the oven to 375°F and grease a 9-by-13-inch baking dish.
Brown the sausage in a skillet over medium-high heat for 7 to 9 minutes, breaking it into crumbles, until no pink remains and it reaches 160°F; drain it well.
Unroll the crescent dough into the dish and press the seams and edges together to form one solid crust.
Bake the empty crust for 8 minutes, until it looks dry but remains pale, then remove it from the oven.
Scatter the sausage and 1 1/2 cups cheddar evenly over the dough.
Whisk the eggs, milk, salt and pepper until no streaks remain and pour the mixture slowly over the sausage.
Top with the remaining cheddar and bake for 24 to 29 minutes, until the center reaches 160°F and barely jiggles when nudged.
Rest for 10 minutes before cutting into squares.
