Preheat the oven to 350 degrees . Grease a 9-by-13-inch baking dish. In a medium saucepan, bring 3 1/2 cups water and 1/2 teaspoon salt to a boil. Stir in the grits, lower the heat and cook, stirring, until very thick, about 5 minutes. Stir in the cheese and butter until melted. Transfer to the prepared baking dish and smooth with a spatula.
Using the back of a spoon, make 8 evenly spaced wells in the grits. Crack 1 egg at a time into a small bowl and pour into each well. Bake the casserole until the egg whites are set but the yolks are still jiggly, about 20 minutes.
Meanwhile, in a large skillet, heat the olive oil over medium-high heat. Add the bell peppers and onions and cook, stirring frequently, until the onions are browned, about 10 minutes; season with salt and pepper. Lower the heat to low and cook until the peppers are softened, about 5 minutes. Serve the grits casserole with the peppers and onions; sprinkle with hot sauce.