Combine mashed potatoes, egg, and pepper in a bowl and mix well together.
Pour bread crumbs into a small bowl. Shape potato mixture into 3-inch patties with wet hands and dip in bread crumbs so they are evenly coated.
Heat oil and butter in a nonstick skillet over medium heat and cook potato cakes on both sides until they are heated through and browned, about 4 minutes per side.