Place potatoes into a saucepan and fill with enough water to cover. Bring to a boil, then simmer over medium heat until tender enough to pierce with a fork, about 10 minutes. Drain and mash potatoes, then set aside.

Meanwhile, combine cabbage and carrot in another saucepan and add just enough water to barely cover the bottom of the pan. Cook over medium heat until tender, about 5 minutes. Drain off liquid and stir into mashed potatoes along with onion, mustard, and Cheddar cheese. Season with salt and pepper.

Shape mixture into 4 patties. Heat a greased grill pan over medium-high heat. Place patties on the pan and grill until heated through and golden brown, about 5 minutes on each side. Serve right away.