Combine 3 tablespoons vinegar, oil, and next 3 ingredients in a shallow dish or large zip-top plastic freezer bag; add mushrooms, turning to coat. Cover or seal, and chill 1 hour, turning occasionally. Remove mushrooms from marinade, discarding marinade.
Grill mushrooms, covered with grill lid, over medium-high heat (350° to 400°) 3 to 4 minutes on each side or until tender. Remove mushrooms from grill, and immediately sprinkle undersides evenly with blue cheese.
Stir together mayonnaise and 1 teaspoon vinegar. Spread mixture evenly on cut sides of buns; place lettuce, mushrooms, and tomato slices on bottom halves of buns, and cover with tops. Serve immediately.