In a large heavy-bottomed saucepan, combine 2 cups water and the milk and bring to a boil over medium-high heat. Stir in the grits and return to a boil. Season with 1 teaspoon of salt and pepper. Decrease the heat to low and simmer, stirring occasionally, until creamy and thick, 45 to 60 minutes.
Meanwhile, heat a medium skillet over medium heat. Add the sausage and cook, stirring occasionally to break up the sausage, until no longer pink and the meat is cooked through, 5 to 7 minutes. Remove with a slotted spoon to a paper-towel-lined plate to drain and cool slightly.
Remove the grits from the heat and add the cheese and butter. Taste and adjust for seasoning with salt and pepper and stir to combine. Add the reserved sausage, cayenne, eggs, green onions and jalapenos and stir until well incorporated.
Butter a large (9-by-13-inch) casserole dish. Spoon the mixture into the prepared dish. At this point, the casserole can be covered and refrigerated overnight (bring to room temperature before baking), or baked immediately.
Bake at 350 degrees F from room temperature until bubbly and golden brown, about 1 hour. Remove to a rack to cool slightly before serving.