In small saucepan, boil potatoes til tender, drain, set aside.
Meanwhile, in a large saucepan, heat the butter over medium heat.
Add onion, carrot and shallots; reduce heat to medium-low.
Cook, stirring frequently, for about 10-15 minutes, or until onion is very soft and golden brown. Add garlic during last 5 minutes.
In a large screwtop jar, mix broth and flour, shaking until smooth.
Stir into onion mixture.
Add milk, caraway seeds, and pepper. Cook over medium heat 5 minutes, or until thickened, stirring frequently.
Gradually stir in cheese, mustard and horseradish, reduce heat to low.
Cook and stir until cheese melts, but do not boil.
Stir in brats, 12 oz. beer, and potatoes.
Cook, stirring frequently, until heated through. *You can add a little of the beer from cooking the brats if desired. See note below.
Garnish with additional shredded cheese, if desired.
Garnish with a sprinkling of cayenne or paprika if desired.
Makes 4-6 main-dish servings.
*Note: When cooking the brats: First brown in pan with just a bit of oil, then add 1 cup beer and simmer til done. Remove brats from beer and cut and slice. *Save beer broth to add to pot, as much or little as you like, for thickness.