Preheat the oven. Set the oven to 325°F (163°C). Lightly grease a casserole dish and set aside.
Brown the sausage. Place the pork sausage in a heavy skillet over medium heat. Cook, breaking it into small crumbles with a spatula, until browned and fully cooked. Drain the excess fat and set aside.
Prepare the grits. Bring 2¼ cups of water and 1 teaspoon salt to a boil in a saucepan. Gradually stir in the quick-cooking grits.
Cook until thickened. Cover the saucepan, reduce the heat to low, and cook for approximately 5 minutes, stirring occasionally, until the grits become thick and tender.
Start the cheese sauce. In a separate medium saucepan, melt the butter or margarine over medium-low heat. Stir in the flour and black pepper until combined.
Add the milk. Gradually pour in the milk while stirring continuously. Continue cooking until the sauce becomes smooth and thick enough to lightly coat a spoon.
Melt the cheese. Add the shredded cheddar cheese and stir until melted and fully incorporated into the sauce.
Combine the sausage and grits. Add the cooked sausage and half of the prepared cheese sauce to the grits. Stir until everything is evenly distributed.
Transfer to the casserole dish. Spoon the cheesy sausage and grits mixture into the prepared baking dish and spread it into an even layer.
Add the eggs. Use the back of a large spoon to create four indentations in the grits. Carefully crack one egg into each indentation.
Bake the casserole. Place the dish in the preheated oven and bake for approximately 15–18 minutes, or until the eggs reach your preferred doneness. For fully cooked eggs, continue baking until the yolks and whites are firm and the egg portions reach 160°F (71°C).
Warm the remaining sauce. While the casserole finishes baking, gently reheat the reserved cheese sauce over low heat, stirring until smooth.
Serve warm. Divide the breakfast bake into four portions and serve with the warm cheese sauce spooned over or alongside each serving.