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.