Preheat the oven to 375 degrees F (190 degrees C).
Heat a large skillet over medium-high heat. Cook and stir sausage in the hot skillet until browned and crumbly, 5 to 7 minutes. Remove sausage to a plate; drain and discard most of the grease from the skillet. Add hash browns and cook until golden brown, 5 to 7 minutes.
Beat eggs, milk, salt, and pepper together in a bowl.
Arrange crescent rolls on the bottom of a 9x13-inch baking dish. Layer with sausage and hash brown potatoes. Pour egg mixture over top and sprinkle with pepper Jack cheese.
Bake in the preheated oven until a knife inserted into the center comes out clean, about 30 minutes. Remove from the oven and let set before serving, about 10 minutes.