How to Make It
Lightly oil a jelly-roll pan or other shallow-rimmed baking sheet. In a bowl, combine the cornmeal with 2 cups cold water and set aside to soak.
Put the sweet potatoes in a medium saucepan, cover with cold water and bring to a boil. Boil the sweet potatoes until they are very soft, about 20 minutes. Drain well and pass them through the fine disk of a food mill. Alternatively, mash the sweet potatoes by hand until smooth. (You should have about 1 1/4 cups.)
In another medium saucepan, bring 2 cups of water to a boil. Add 2 tablespoons of the butter, 1 teaspoon salt, 1/2 teaspoon pepper and the cornmeal mixture. Bring to a boil, reduce the heat to moderately high and cook, stirring often, until the mixture is very thick, 6 to 8 minutes. Remove from the heat and stir in the sweet potato puree. Season with salt and pepper. Spread the mixture on the prepared baking sheet in an even 2/4-inch-thick layer and cover with wax paper. Refrigerate until firm, at leas 1 hour.
Using a round or oval 2- to 3-inch fluted cookie cutter, stamp out the galettes. Alternatively, the sweet potato mixture can be cut into 2-inch squares with a knife.
Preheat the broiler. In a large skillet, melt the remaining 2 tablespoons butter over moderately high heat. Fry the galettes in batches until lightly browned on both sides. As they are done, transfer them to cookie sheets. Sprinkle with the cheese and broil 4 to 6 inches from the heat for about 3 minutes, or until lightly browned around the edges and heated through. Serve immediately.