Rinse the mushrooms and pat dry with paper towels. Remove the thick stems, then scrape out the dark gills with a spoon. Place the cleaned mushrooms in a large zip-top plastic bag. In a medium bowl, whisk together the olive oil, balsamic vinegar, tamari, mustard, and vegan Worcestershire sauce until well-combined. Pour the marinade over the mushrooms, squeeze any excess air out of the bag, and close tightly. Place the bag in a shallow baking dish, then marinate at room temperature for 30 minutes. For a fuller flavor, place the mushrooms in the refrigerator and allow to marinate overnight, turning once or twice if you think of it.
Drain off the marinade (you can reserve it for another batch of burgers if you like), and grill the mushroom caps hollow-side down for 5 to 7 minutes, then flip and grill for an additional 4 to 7 minutes, until juicy and tender. Serve on buns or lettuce wraps with your favorite burger fixings.