Heat the oven to 425°F. Set a rack in the upper middle position and place a large pot, a colander, and a 10-inch oven-safe skillet near the stove.

Put the potatoes into the pot. Cover them with cold water by about 1 inch and add 1 tablespoon of the kosher salt.

Set the pot over high heat and bring the water to a boil. Lower the heat just enough to keep a steady boil and cook the potatoes for 12 to 15 minutes, until a fork slides in easily.

While the potatoes cook, pour the olive oil into the skillet and set it over medium heat. Add the onion and 1/2 teaspoon of the black pepper.

Cook the onion for 4 to 5 minutes, stirring often, until it softens and starts to turn lightly golden at the edges.

Add the cabbage to the skillet with 1/2 teaspoon of the kosher salt. Cook for 5 to 6 minutes, stirring often, until the cabbage shrinks down and begins to soften.

Add the garlic, tomato paste, smoked paprika, and dried thyme to the skillet. Stir for 1 minute, until the tomato paste darkens slightly and smells rich.

Add the cannellini beans and vegetable broth to the skillet. Stir well and scrape along the bottom of the skillet so the tomato paste mixes evenly into the liquid.

Bring the filling to a gentle simmer. Cook for 6 to 8 minutes, stirring once or twice, until the cabbage is tender and the liquid has reduced to a thick, spoonable sauce.

Taste the filling and add more kosher salt or black pepper if needed. Spread the filling into an even layer in the skillet and take the skillet off the heat.

Drain the potatoes in the colander. Return the potatoes to the pot and let them sit off the heat for 1 minute so extra steam can escape.

Add the milk and butter to the pot with the potatoes. Mash until mostly smooth, then stir in 1/2 cup of the cheddar and the remaining 1/2 teaspoon kosher salt.