barbecued pork burgers with slaw

1 (1/2-pound) piece green cabbage, cored
1/4 cup mayonnaise
1 tablespoon milk
1 tablespoon plus 1 1/2 teaspoons white-wine vinegar, divided
1/2 cup very finely shredded carrot (1 medium)
1 tablespoon thinly sliced fresh chives
1/2 cup bottled tomato-based barbecue sauce
1/4 teaspoon cayenne
1 1/2 pounds ground pork
4 kaiser or soft rolls, split and grilled

Prepare grill for direct-heat cooking over medium-hot charcoal (medium heat for gas). Thinly slice enough cabbage with slicer to measure 2 cups. Whisk together mayonnaise, milk, and 1 1/2 teaspoons vinegar until smooth, then toss with cabbage, carrots, chives, and salt and pepper to taste. Let coleslaw stand at room temperature, uncovered, while making burgers. Stir together barbecue sauce, cayenne, 1/4 teaspoon salt, and remaining tablespoon vinegar until combined well. Mix together pork, 1/2 teaspoon salt, 1/4 teaspoon pepper, and 2 tablespoons barbecue sauce mixture until combined (do not overmix), then form into 4 (3/4-inch-thick) burgers (4 inches in diameter). Oil grill rack, then grill patties, covered only if using a gas grill, turning over occasionally, until just cooked through, about 6 minutes total. Brush top of each patty with 1 tablespoon barbecue sauce mixture, then turn over and grill 30 seconds. Brush top of each patty with 1 tablespoon barbecue sauce, then turn over and grill 30 seconds more. Brush cut sides of rolls with remaining 1/4 cup barbecue sauce, then sandwich patties and coleslaw between rolls.