Heat the oven to 350°F and butter a 2 1/2-quart baking dish. Bring the water, milk and salt to a gentle boil in a saucepan.
Whisk in the grits, reduce to low and cook for 5 to 7 minutes, stirring often, until tender and thick. Use the package timing if your quick grits need longer.
Brown the diced sausage in a skillet for 5 minutes. Stir it into the cooked grits with the butter, cheddar, pepper and paprika.
Cool for 10 minutes, stirring twice. Stir 1 cup warm grits gradually into the beaten eggs, then fold that mixture back into the pot.
Transfer to the dish and bake for 35 to 40 minutes, until browned at the edges and the center reaches 165°F. Rest for 10 minutes before spooning out portions.