Heat a grill pan on medium-high (or use regular grill). Lightly brush mushroom caps and onion with oil. Grill both for 4 to 5 minutes per side or until mushrooms are tender. (NOTE: Mushrooms will begin to release their juices when they are tender.) Set aside on separate plate. Grill buns, cut-sides-down, for 1 to 2 minutes per side or until toasted.
Meanwhile, prepare sauce: Combine all sauce ingredients in a small bowl. Spread sauce on bottom half of each bun, dividing evenly. Top with onion, mushroom caps, cheese and lettuce.