In a small saucepan, combine soy sauce, mirin, rice vinegar, minced ginger, and sugar.

Simmer the mixture, stirring until the sugar dissolves.

Continue simmering until the marinade reduces to about 1/2 cup.

Allow the marinade to cool to room temperature.

Place the Portobello mushroom caps in a large sealable plastic bag.

Pour the cooled marinade over the mushrooms, arranging them in a single layer.

Seal the bag, pressing out any excess air.

Marinate the mushrooms at room temperature for 1 hour, turning the bag occasionally.

In a bowl, whisk together mayonnaise, rice vinegar, sesame oil, and honey.

Add the finely shredded Napa cabbage, carrot, and chopped scallions to the bowl.

Toss the vegetables well to coat them in the dressing.

Season the slaw with salt and pepper to taste.

Preheat the broiler.

Remove the mushrooms from the marinade, discarding the marinade.

Arrange the mushrooms, stemmed sides up, on a lightly oiled rack of a broiler pan.

Broil the mushrooms 2 inches from the heat until lightly browned.

Turn the mushrooms over.

Brush the mushrooms lightly with vegetable oil.

Broil for 3 minutes, or until the mushrooms are tender.

Transfer the cooked mushrooms with a slotted spatula to the bottoms of the toasted sesame-seed hamburger buns.

Top each mushroom with the Napa cabbage slaw.

Place the bun tops on the slaw and serve immediately.