Prepare the mushrooms by removing the stems and cleaning the caps with a damp cloth or paper towel. Place the mushrooms in a rimmed plate and drizzle with olive oil, balsamic vinegar, and tamari. Sprinkle with salt and several grinds of pepper. Use your hands to rub the seasonings into the mushrooms, coating them on both sides.

Heat a grill or grill pan to medium heat. Place the mushrooms, gill side up, onto the grill. Cook for 5 to 7 minutes per side, or until the mushrooms are tender.

Serve the mushroom burgers on the hamburger buns with desired fixings.