Brown the sausage: In a large skillet over medium heat, cook the sausage, breaking it into crumbles with a wooden spoon. Cook until no longer pink, about 5-7 minutes. Transfer to a paper towel-lined plate to drain, reserving about 1 tablespoon of drippings in the pan.
Sauté aromatics: In the same skillet with the drippings, add green onions (white parts) and garlic. Sauté until fragrant, about 1 minute.
Cook the grits: In a medium pot, bring the chicken broth and milk to a gentle boil. Slowly whisk in the grits, stirring constantly to prevent lumps. Reduce heat to low and cook, stirring frequently, for about 15-20 minutes until thick and creamy. (Stone-ground grits take longer than quick-cooking varieties, so adjust time accordingly.)
Add the good stuff: Once the grits are tender, stir in the butter until melted. Remove from heat and add 1½ cups of the cheddar cheese, stirring until completely melted and incorporated.
Combine everything: Fold in the cooked sausage, sautéed green onions and garlic, black pepper, and cayenne if using. Taste and add salt as needed (the amount will depend on how salty your broth and sausage are).
Serve it up: Spoon into bowls, top with remaining cheese and green onion tops. Serve immediately while hot and creamy.