Heat oven to 350°F; butter 2 1/2-quart dish. Gently boil water, milk, and salt in saucepan.
Whisk in grits; reduce to low and cook 5 to 7 minutes, stirring often, until tender and thick. Follow package if longer cooking needed.
Brown sausage in skillet 5 minutes. Stir into cooked grits with butter, cheddar, pepper, and paprika.
Cool 10 minutes, stirring twice. Gradually stir 1 cup warm grits into beaten eggs; fold back into pot.
Transfer to dish; bake 35 to 40 minutes until edges browned and center 165°F. Rest 10 minutes before spooning portions.