Bratwurst and Red Cabbage

Name: Bratwurst and Red Cabbage
Ingredients: 1 pound uncured bratwurst; 2 tablespoons olive oil; 1 12-ounce bottle Pilsner or other lager, divided; 1 medium red onion, chopped; 1/2 medium head of red cabbage, thinly sliced; 1 medium red beet, peeled, coarsely grated; Kosher salt, freshly ground pepper; 1/2 cup apple cider vinegar; 1 tablespoon light brown sugar; 1/4 teaspoons ground allspice; Freshly grated horseradish (for serving)