Preheat the oven to 350 degrees F. Butter a 9 by 13-inch baking dish with 1 tablespoon butter. Add 3 tablespoons cornmeal to the dish and tilt the dish to coat the bottom and sides with the cornmeal. Set the baking dish aside.
Combine the milk, cream, buttermilk, salt, and pepper in a medium saucepan, and bring just to a boil. Whisk in the 1 cup cornmeal and cook, stirring constantly, until the mixture thickens and has the consistency of grits, 1 to 2 minutes. Remove the saucepan from the heat and immediately transfer the mixture to a heatproof bowl. Stir in the cheese, and set the bowl aside until the mixture is lukewarm, stirring often to prevent a skin from forming on the top, 10 to 15 minutes.
In a small bowl, lightly beat the egg yolks. Then stir the yolks, green onions, thyme, baking soda, and baking powder into the cornmeal mixture.
Combine the egg whites and the sugar in a mixing bowl, and beat until stiff peaks form. Fold 1/3 of the whites into the cornmeal mixture to lighten it. Then gently fold in the remaining whites, taking care not to deflate the whites.
Transfer the mixture to the prepared baking dish, and bake until the spoonbread is puffed and golden brown on top, and a knife inserted into the center comes out clean, about 30 minutes. Serve immediately.