In a large skillet, sauté the sausage until fully cooked, breaking into little pieces as you cook the sausage.
Add butter. Once melted add in the flour and stir until well incorporated. Add the milk, stir.
Turn heat down and allow to continue cooking until the gravy thickens. Remove from heat.
Cover the baking dish with parchment paper or grease with cooking spray. Lay out the crescents into a 9x13 casserole dish. Pour the sausage gravy over the crescent layer.
Whisk together eggs and 1 Tbsp milk with salt and pepper. Pour over the sausage gravy layer.
Sprinkle the shredded cheese over the egg mixture. Cover with foil.
Bake at 425°F for 25 minutes, remove foil and bake another 5 minutes.