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.