Make teriyaki marinade:

In a small saucepan simmer marinade ingredients, stirring until sugar is dissolved. Continue simmering marinade until reduced to about 1/2 cup and cool to room temperature.

When the marinade is cooled put the mushroom caps and marinade in a large sealable plastic bag, arranging caps in one layer, and seal bag, pressing out excess air.

Marinate mushrooms at room temperature, turning bag over several times for approx. 1 hour. While the mushrooms are marinating make the slaw.

In a bowl combine the mayonnaise, vinegar, oil, and honey. Whisk them together and add the cabbage, carrot, and scallions. Toss vegetables well to coat and season with salt and pepper.

Preheat the grill or broiler. Drain mushrooms and arrange, stemmed sides up, on lightly oiled grill or broiler pan.

Broil mushrooms 2 inches from heat and turn over. Brush mushrooms lightly with oil and broil 3 minutes, or until tender.

Transfer mushrooms with a slotted spatula to hamburger bun bottoms. Top mushrooms with slaw and bun tops.