Make the coleslaw: In a large bowl, whisk the mayonnaise with the onion, vinegar, sugar and celery seeds. Add the shredded cabbage and carrot, season with salt and pepper and toss well. Let stand until slightly softened, about 25 minutes.
Make the burgers: Light a grill. Form the beef into 4 patties; rub with oil and season with salt and pepper. Sear over high heat, turning once, for 5 minutes for medium-rate Brush with 1/2 cup of the barbecue sauce and grill until glazed, about one minute per side.
Grill the buns and brush them with the remaining barbecue sauce. Top with burgers, picks and coleslaw and serve.