Preheat the oven to 350°F. Butter a large casserole.
In a large heavy-bottomed saucepan, combine the water and milk and bring to a boil over medium-high heat. Stir in the grits and return to a boil. Season with salt and pepper. Decrease the heat to low, and simmer until creamy and thick, 45 to 60 minutes. Remove from the heat. Add the cheese and 2 tablespoons butter.
Taste and adjust for seasoning with salt and pepper and stir to combine. Add the eggs, cayenne, jalapeño, and green onions and
stir until well incorporated. Spoon the mixture into the prepared casserole. Bake until bubbly and golden brown, about 1 hour. Remove to a rack to cool slightly before serving.