Cook the sausage in a skillet over medium-high heat until cooked through. Remove the sausage to a paper towel to drain. Use the rendered sausage fat to grease 4 large individual ramekins, (gratin dishes or individual cast iron pots work too) and arrange them on a rimmed sheet tray.
In a large saucepan, over medium heat, bring the water and half-and-half to a boil and season with salt, to taste. Rapidly whisk in the grits. Continue whisking while returning to a boil, about 5 to 7 minutes. Reduce the heat to a simmer and cook until thick and creamy, stirring frequently, about 20 minutes. Season with pepper and more salt, if needed.
Divide the cooked grits between the prepared baking dishes and top each with cooked sausage and grated cheese. Make a divot with the back of a spoon into the cheese and sausage topped grits. Crack the eggs, 1 at a time, into a small bowl and top the ramekins with the eggs. Bake until the egg whites are almost, but not fully set, about 10 to 15 minutes. For more well-done eggs, bake until the whites are set.