Put oven rack in middle position and preheat oven to 375°F. Generously butter bottom and sides of a 13- by 9-inch baking dish.
Cook sausage in a 12-inch heavy skillet over moderately high heat, stirring frequently and breaking up any large lumps with a fork, until browned, about 10 minutes. Pour off fat from skillet, then cool sausage to room temperature.
Cut half of loaf into 1-inch-thick slices and reserve remaining half for another use. Pulse butter and garlic in a food processor until smooth. Spread a thin layer of garlic butter on both sides of each bread slice, arranging bread in 1 layer in bottom of baking dish. Sprinkle sausage on top.
Whisk together eggs, milk, salt, and pepper in a large bowl until frothy, then whisk in scallions and half of cheese. Pour egg mixture over sausage (bread will float to the top), pushing down on bread with a spatula to help it absorb liquid. Sprinkle with remaining cheese.
Bake, covered with a large sheet of buttered foil (buttered side down), 30 minutes, then carefully remove foil and bake until top is slightly puffed and eggs are cooked through in center, about 20 minutes more.
Let stand 10 minutes, then cut into 12 squares.
Save time by using store-bought garlic bread.
Substitute ham for the sausage, if desired.
Prepare as directed and freeze for up to 2 months before baking.