Beer-Can Cabbage Sandwiches

1) Prepare a grill for medium-high heat 2) Cut a 3x3" square around core of cabbage at the base 3) Carefully remove core with knife 4) Using knife and a spoon, create a 3"–deep cavity in cabbage, big enough to hold a beer can 5) Discard core but reserve leftover cabbage from hollowing out cavity 6) Using a brush, coat cabbage with oil, then season generously with salt and pepper 7) Pour out (or drink) half of the beer 8) Place beer can in cavity of cabbage so cabbage sits upright 9) Transfer to grill 10) Using brush, generously coat cabbage with 3 Tbsp 11) barbecue sauce 12) Cover grill and cook cabbage, brushing with sauce every 15 minutes, until outer leaves are crispy and dark brown and cabbage is cooked through and tender when pierced with a paring knife, 45–50 minutes total 13) During the last 5 minutes of cooking, grill white onion rings until charred 14) Meanwhile, whisk mayonnaise, vinegar, honey, and remaining 1/2 tsp 15) salt and 1/4 tsp 16) pepper in a medium bowl 17) Thinly slice reserved cabbage 18) Add to bowl with dressing along with carrot and red onion; toss to combine 19) Remove cabbage from grill and discard can 20) Let cool slightly 21) Cut grilled cabbage in half lengthwise, then slice each half crosswise into 3/4" strips 22) Coarsely chop white onion rings and toss with cabbage in a medium bowl with remaining 1 1/4 cups barbecue sauce (you should have about 6 cups barbecued cabbage) 23) Halve rolls and grill until toasted, about 30 seconds 24) Arrange 1 cup barbecued cabbage on each bottom bun 25) Top each with 2 slices cheese, 1/4 cup cabbage slaw, and pickled jalapeños, if using 26) Top with top buns 27) Barbecued cabbage and coleslaw can be made 3 days ahead; cover separately and chill 28) Reheat gently in a covered pot on the stovetop or in the microwave