2 cups stone-ground grits
1/2 teaspoon salt, plus more if needed
4 large eggs
12 tablespoons (1 1/2 sticks) unsalted butter, cut into tablespoon pieces, plus more for buttering the dish
3 cups grated sharp Cheddar (or other cheese), plus more if needed
1/2 teaspoon cayenne pepper, plus more if needed
4 cloves garlic, finely chopped