In a saucepan over medium heat, add milk, water, and salt. Bring to a boil, then slowly whisk in the grits.
Reduce heat to medium-low and cook for 20 minutes, whisking frequently. If grits begin to stick to pan, reduce heat and add a splash of water. Grits should be creamy and thick, with no rough texture remaining when they're done cooking.
Remove from heat and add pepper, butter, and shredded cheese. Stir until melted and set aside.
While grits are cooking, warm a non-stick skillet over medium heat. Crumble sausage into pan and cook until browned on all sides, turning occasionally with a spatula, about 6-8 minutes. Transfer cooked sausage out of pan, draining all but 1 tablespoon of grease.
Crack eggs into a bowl and whisk until combined. Pour eggs into your greased non-stick skillet over medium to medium-low heat. As eggs begin to set on the bottom of the pan, pull the eggs across the pan with a spatula. This will form egg curds in the pan. Continue pulling and folding the eggs until no liquid remains. Remove from heat.
Divide cheesy grits between four bowls, then top each bowl with sausage and scrambled eggs. Optionally, add more shredded cheese on top. Garnish with chopped chives and serve immediately.
Leftover grits bowls will keep for 3-4 days in a sealed container in the refrigerator. Microwave at medium heat, adding a splash of water as needed to thin your grits.
Sausage can be pork, chicken, turkey, mild, or hot sausage. Or, substitute with cooked and crumbled bacon.
If using instant grits, follow the package directions instead of my directions above. Instant grits will cook much more quickly.