In a large saucepan, heat the butter over medium heat. Add onion, carrot and shallots; reduce heat to medium low. Cook, stirring frequently, about 10 to 15 minutes or until the onion is very soft and golden.
In a large screw-top jar, combine broth and flour. Cover and shake until combined and smooth. Stir into the onion mixture. Add the milk, caraway seeds and black pepper. Cook over medium heat, stirring frequently, about 5 minutes or until the mixture thickens. Gradually stir in the cheese; reduce heat to low. Cook, stirring frequently, until cheese melts, but do not boil. Stir in the bratwurst and beer. Cook, stirring frequently, until heated through. If you like, serve with rye bread. Makes 4 to 6 main-dish servings (7 cups).