Hollow out the red cabbage and set the stalk aside for now.
Pour out half of the beer from the can and stick the cabbage head on it.
Stand the beer can upright on the grill and baste the cabbage head with BBQ sauce. Grill the cabbage for about 30-50 minutes and brush it with BBQ sauce again after 15 minutes. The cooking time will depend on the size of the cabbage.
Take the cabbage off the grill and remove the beer can. Dice the red cabbage and mix it with the BBQ sauce.
For the coleslaw, finely dice the red onion and the cabbage stalk you set aside earlier. Add the shredded carrot, mayonnaise, apple cider vinegar, and honey, season with salt and pepper, and mix well.
Spread a layer of the BBQ beer slaw on the bottom half of each burger bun and top with a hamburger patty, cheddar cheese, sliced jalapeños, and 1-2 tablespoons of the homemade coleslaw for each burger. Put the top burger buns on and dig in!
