Preheat the oven to 400 degrees F.

Butter an 8-cup gratin dish or souffle dish with 2 teaspoons of the butter and set aside.

In a large heavy saucepan, bring the milk and remaining butter just to a boil. Whisk in the cornmeal, little by little, until smooth and thoroughly combined. Return mixture to a boil, reduce heat slightly so the mixture remains at a low boil, partially cover pan and cook until quite thick, 4 to 5 minutes. Transfer to a large mixing bowl, add the green onions, salt and pepper and hot sauce, to taste. Let stand, covered with plastic wrap, for 10 to 15 minutes.

In a separate bowl, whisk the eggs until light and fluffy. Whisk the eggs into the hot cornmeal mixture, along with 1 cup of the Cheddar. Turn mixture into the buttered dish and place entire dish on a baking sheet. Sprinkle the top with the remaining 1/4 cup of cheese. Bake until firm, puffy, and golden brown on top, 35 to 40 minutes. Serve immediately.