Preheat the oven to 350 degrees F. Grease a 4-quart casserole dish.
Bring the broth, salt, and pepper, to a boil in a 2-quart saucepan. Stir in the grits and whisk until completely combined. Reduce the heat to low and simmer until the grits are thick, about 8 minutes. Add the garlic-cheese and milk and stir. Gradually stir in the eggs and butter, stirring until all are combined. Pour the mixture into the prepared casserole dish. Sprinkle with the Cheddar and bake for 45 minutes, or until set.