Preheat the broiler. On a baking sheet, broil the English muffin halves, cut side up, 4 inches from the heat for about 1 minute, or until golden. Turn the oven down to 375°. Butter the halves, stack and cut them into 1-inch cubes. Arrange the cubes in an even layer in four 8-ounce gratin dishes or other small shallow baking dishes.
In a small nonstick skillet, heat the oil. Add the breakfast sausage, breaking it up with the back of a spoon, and cook over moderately high heat until browned and cooked through, about 3 minutes. Drain the sausage on paper towels and divide it among the gratin dishes.
Crack 2 eggs into each gratin dish and sprinkle with the cheddar. Bake for 17 minutes, or until the egg whites are firm and the yolks are still runny. Serve immediately, with hot sauce on the side.