Place mushrooms into a large resealable plastic bag; add soy sauce, balsamic vinegar, and garlic. Seal bag and shake to combine ingredients; marinate mushrooms at room temperature for 2 to 3 hours, turning bag occasionally.

Preheat grill for medium-high heat and lightly oil the grate.

Place red bell pepper onto the grate and grill until the skin is blistered and charred, turning often, about 10 minutes. Transfer pepper to a bowl and cover with plastic wrap; let pepper cool. When pepper is cool enough to handle, remove and discard skin, stem and seed the pepper, and chop.

Mix pepper into mayonnaise in a bowl and stir in cayenne pepper. Chill red pepper mayonnaise until serving time.

Grill marinated mushroom caps until browned and juicy, about 6 minutes per side. Serve mushrooms on buns, topped with red pepper mayonnaise.