In a medium saucepan, combine the water or broth, milk, salt, and cayenne and bring to a boil. Slowly whisk in the grits and lower the heat to a simmer. Cover and cook, stirring occasionally, until thickened, about 30 minutes.
Stir in the cheese and butter until evenly incorporated. Season with pepper to taste. Serve immediately.
Remove from the oven and let cool for 5 minutes before serving.
For Baked Grits: Preheat the oven to 350 degrees F. Cool the grits slightly, stir in the 3 beaten eggs. Transfer grits to a buttered deep casserole dish, smooth the top with a rubber spatula, and bake until just set and lightly browned, about 1 hour.
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