Heat the oil in a large skillet, preferably cast iron, over medium-high heat. Add the shallot and cook for 1 minute. Add the sweet potato and cook, stirring occasionally, for 4 minutes. Add the Brussels sprouts and cook, stirring occasionally, for another 4-5 minutes, or until the vegetables are cooked through and browned.

Using the back of a large spoon, create 4 indentations in the hash. Break the eggs into the indentations and cover the pan. Cook for another 2-3 minutes or until the whites of the eggs are cooked through but the yolks are still soft. Sprinkle with a little salt and pepper, add the cheese, and serve right from the skillet.