Prepare grill; set to medium heat.
To prepare cabbage, heat oil in a large nonstick skillet over medium heat. Add onion; cook 5 minutes or until tender, stirring occasionally. Add cabbage and next 5 ingredients (cabbage through pepper). Cover and cook 5 minutes; stir well. Cook an additional 15 minutes or until cabbage is tender, stirring occasionally. Remove from heat; keep warm.
To prepare duck, brush meat side of breasts with barbecue sauce; place on grill rack, skin side up. Cook 8 minutes. Turn duck over; brush with remaining barbecue sauce. Cook 8 minutes or until desired degree of doneness; remove from heat, and let stand 5 minutes. Carefully remove skin from breasts; cut each breast in half. Cut each breast half into 1/2-inch-thick slices. Spoon cabbage onto plates; arrange duck around cabbage.