How to Make It
Heat oven to 450° F. Toss the potatoes with 1 tablespoon of the oil and ¼ teaspoon each salt and pepper on a rimmed baking sheet. Roast for 5 minutes. Add the bratwurst to the baking sheet and continue to roast, turning the potatoes once, until the potatoes are golden and the bratwurst is cooked through, 12 to 15 minutes more.
Transfer the potatoes to a medium bowl and let cool slightly. Add the cucumber, yogurt, dill, and ¼ teaspoon each salt and pepper and stir to combine.
Meanwhile, heat the remaining tablespoon of oil in a medium pot over medium-high heat. Add the cabbage, onion, caraway seeds (if desired), and ¼ teaspoon each salt and pepper. Cook, stirring occasionally, until the cabbage and onion are tender, 6 to 8 minutes. Stir in the vinegar.
Place the bratwurst in the buns. Top with the mustard (if desired) and sautéed cabbage, dividing evenly. Serve with the potato salad.