In a medium-size saucepan over high heat, bring 2 1/2 cups water to a boil. Add salt, reduce heat to medium and slowly whisk in grits. Cook, uncovered, stirring frequently, until grits are tender and thickened, about 20 minutes.
Using a sharp paring knife, cut off ends of sausage casings. Squeeze out sausage meat and discard casings. Cook sausage in a medium-size frying pan over medium-high heat, stirring occasionally with a wooden spoon (to crumble large pieces) until fully cooked, about 5 minutes. Remove sausage from pan with a slotted spoon and set aside. Leave accumulated fat in pan.
Remove grits from heat and stir in cheese and cooked sausage.
Return frying pan with fat to medium-high heat. Carefully crack eggs into frying pan one at a time. Cook until whites are set but yolks are still runny, about 3 minutes. (For firmer yolks, flip eggs and cook for 1 minute longer.) Spoon grits mixture onto individual plates and top each serving with an egg. Serve immediately.