Rinse and drain the sauerkraut, then squeeze out excess brine.
Slice the sausage, halve the potatoes, slice the onion, and dice the apple. Melt the butter.
Whisk broth, melted butter, brown sugar, mustard, garlic, caraway, pepper, and salt.
Add potatoes and onion to the slow cooker; pour in half the sauce and toss.
Spread sauerkraut and apple over potatoes. Add sausage on top. Pour in remaining sauce.
Cover and cook on Low for 6 hours, or on High for about 3 hours.
Stir gently, taste, and adjust salt or pepper. Serve hot with mustard or rye.