Preheat the oven to 350°F.
Place the potatoes in a large pot. Add enough cold water just to cover the potatoes, then add the salt. Cover and bring to a boil over medium-high heat. Once boiling, set a timer for 15 minutes. When the timer goes off, add the cabbage to the pot with the potatoes and let cook for another 8 minutes.
Drain well in a colander, then add them back to the pot you boiled them in.
Add the butter to the pot with the vegetables and mash roughly. You’re not looking for a smooth mixture, you still want some chunks for texture. Season with salt and pepper to taste, then transfer to an 8-inch square baking dish (you can also use a 9-inch round baking dish).
Cover the entire surface with grated cheddar cheese, then bake until the top is golden brown, about 45 minutes. Let rest for 5 minutes before serving.
