For the slaw
1 cup very thinly sliced red cabbage
1/4 cup mayonnaise
3 Tbs. pickle brine (from a jar of pickles, preferably dill)
4 tsp. cider vinegar
2 tsp. granulated sugar
Kosher salt and freshly ground black pepper
2-1/2 cups very thinly sliced green cabbage
1/2 cup grated carrots
1/4 cup finely chopped white onion
For the bratwursts
2 12-oz. bottles lager beer, such as Budweiser
4 uncooked bratwursts
1 cup chopped white onion
4 Tbs. unsalted butter, melted
2 Tbs. spicy brown mustard, such as Gulden’s
Vegetable oil for brushing
4 hoagie rolls, split but not all the way through
Make the slaw
Soak the red cabbage in a bowl of ice water for 15 minutes to prevent its color from bleeding. Drain and pat dry. In a large bowl, whisk together the mayonnaise, pickle brine, vinegar, sugar, 1/4 tsp. salt, and a grind of pepper. Add the red and green cabbage, carrot, and onion and toss well to combine. Refrigerate for 2 to 4 hours.
Make the bratwursts
In a 10- to 11-inch skillet, combine the beer, bratwursts, and onion. Add enough water to barely cover the bratwursts and bring just to a simmer over medium heat. Cover, remove from the heat, and let sit for 10 minutes.
Meanwhile, position a rack about 4 inches from the broiler and heat the broiler on high.
In a small bowl, combine the butter, mustard, and 4 tsp. of the bratwurst-cooking liquid; set aside.
Arrange the bratwursts on a broiler pan and lightly brush them with oil. Broil until goldenbrown, turning once, about 4 minutes total. Transfer to a plate. Put the buns on a baking sheet cut sides up and broil until lightly toasted, about 30 seconds.
Spread the insides of the buns with the butter mixture and put the bratwursts in the rolls. Top with the slaw and serve.