1. In a small bowl, combine broccoli slaw, mayonnaise, vinegar and sugar. Refrigerate until serving.
2. In a medium bowl, combine ground beef, mushrooms, scallions, teriyaki sauce and ginger. Form into 8 patties, about 3 inches in diameter; refrigerate for 1 hour.
3. Heat a stovetop grill pan over medium-high heat. Generously coat with nonstick cooking spray or vegetable oil. Grill pineapple slices for 2 minutes per side. Remove to a plate and keep warm. Coat pan again with nonstick cooking spray or oil and add burgers; grill for 3 minutes. Turn and place a slice of cheese on 4 of the burgers; grill 2 more minutes. Top cheese-covered burgers with remaining burgers.
4. To serve, place 1/4 cup broccoli slaw mixture on each bun bottom and top with double burger, a pineapple slice and top half of bun.