Toss the cabbage with the shallots, vinegar, brown sugar and 1 tablespoon salt in a large microwave-safe bowl. Cover the bowl with plastic wrap and microwave on high for 5 minutes. Stir well and let stand for 15 minutes.
Meanwhile, place the brats on the indirect heat portion of the grill and grill, turning occasionally, until the internal temperature reaches 160 degrees F, 15 to 18 minutes. Move the brats over to the direct side of the grill and cook, turning occasionally, until lightly browned on all sides, about 5 minutes.
While the brats finish grilling, melt the butter in a large skillet placed directly on the grill. Add the caraway seeds and cook, stirring, until fragrant, about 1 minute. Add the sauerkraut and stir until well coated. Stir in the dill if using.
Serve the bratwursts with the sauerkraut and assorted mustards on the side.