Using sharp knife, cut ¼-inch slits, spaced ½ inch apart, in crosshatch pattern on surface (non-cupped side) of mushrooms. Combine oil, vinegar, garlic, salt, and pepper in 1-gallon zipper-lock bag. Add mushrooms, press out air, seal bag, turn bag to coat mushrooms, and let sit for 15 minutes.

Combine feta and tomatoes in bowl. Whisk mayonnaise, basil, and lemon juice together in separate bowl.

3A. For a charcoal grill : Open bottom vent completely. Light large chimney starter filled with charcoal briquettes (6 quarts). When top coals are partially covered with ash, pour evenly over grill. Set cooking grate in place, cover, and open lid vent completely. Heat grill until hot, about 5 minutes.

3B. For a gas grill : Turn all burners to high, cover, and heat grill until hot, about 15 minutes. Leave all burners on high.

Clean and oil cooking grate. Remove mushrooms from marinade, reserving any excess marinade. Place mushrooms cupped side up on grill and cook until mushrooms have released their liquid and are charred on first side, 4 to 6 minutes. Flip mushrooms, brush with reserved marinade, and continue to cook until charred on second side, 3 to 5 minutes.

Transfer mushrooms cupped side up to plate and divide feta mixture evenly among caps, packing it down with your hand. Return mushrooms to grill, feta side up, and cook, covered, until heated through, about 3 minutes.

Stack lettuce leaves together to create 4 lettuce wraps. Serve mushrooms on wraps, topped evenly with sauce.