Set a large lidded skillet over medium heat so it is ready when you start cooking.

Make the cool topping: Put the sour cream and prepared horseradish in a small bowl. Stir them together with a small pinch of salt, then set the bowl aside for serving.

Toast the rye crumbs: Add the butter and 1 tablespoon of the olive oil to the warm skillet.

Put the torn rye bread into the skillet. Cook for 3 to 4 minutes, stirring often, until the bread is crisp and browned in spots.

Transfer the toasted rye pieces to a plate and lightly crush them with the back of a spoon so you have rough crumbs. Leave the skillet on the stove.

Brown the sausage: Add the sausage rounds to the same skillet. Cook for 4 to 5 minutes, turning once, until they are browned on both flat sides and have left some fat in the pan.

Transfer the browned sausage to a clean plate. Leave any fat in the skillet.

Start the vegetables: Add the sliced onion to the skillet. Cook for 3 minutes, stirring now and then, until it softens and starts to turn pale gold.

Add the potato pieces to the skillet. Sprinkle in 3/4 teaspoon of the salt and 1/4 teaspoon of the black pepper, then stir so the potatoes are coated in the oil and onion.

Pour the water into the skillet. Cover with the lid and cook for 8 minutes, adjusting the heat as needed so the water bubbles gently but does not boil hard.

Remove the lid and stir the potatoes. Cook for 4 to 5 minutes more, stirring every minute or so, until the water has cooked away, the potatoes are tender when pierced with a knife, and some edges are browned.

Soften the cabbage: Add the sliced cabbage to the skillet. Pour in the remaining 1 tablespoon olive oil and toss so the cabbage starts to coat in the hot fat.