Heat the butter in a large heavy pot over medium high heat until melted and hot. Add the sausages and brown them, turning once, about 5 minutes. Transfer the sausages to a platter.
Stir in the onion, apple, garlic and ½ teaspoon each salt and pepper, then cook, stirring occasionally, until onions begin to turn golden brown, about 6 minutes.
Place the sausage back in the pot along with the cabbage, beer, Worcestershire, mustard, and caraway and bring the mixture to a boil. Reduce the heat to a simmer, then cook, covered, until cabbage is tender, about 20 minutes. Uncover the pot and simmer until liquid is almost evaporated, about 5 minutes. Season with salt and pepper to taste, sprinkle with the dill and serve.
