Shape pork into four 4-inch patties about 1/2 inch thick. Grill on the rack of an uncovered grill directly over medium-high heat for 5 to 6 minutes per side or until no pink remains (160 degrees F), turning once.
Meanwhile, in a medium bowl combine shredded cabbage with carrot, mayonnaise, and the 1 teaspoon barbecue sauce; set aside.
Add ham and bread slices to grill for 1 to 2 minutes until ham is lightly browned and bread is toasted, turning once.
Serve pork burgers on toast topped with ham, slaw, and the 1/4 cup barbecue sauce.