Score the smooth, non-gill side of the mushroom caps in a 1/2-inch-wide crosshatch pattern, cutting about 1/8-inch deep. This helps the marinade penetrate and prevents the caps from curling on the grill.

Whisk the olive oil, vinegar, garlic, salt, and pepper together in a large bowl. Add the mushroom caps and onion slices, tossing gently to coat every surface, and let them marinate for 15 minutes.

Combine the crumbled feta, roasted red peppers, and sun-dried tomatoes in a small bowl. In a separate small bowl, stir the mayonnaise and basil together until well mixed.

Prepare the grill for medium-high heat. Place the mushrooms scored-side down and the onion slices on the grill; cook until the mushrooms are lightly charred and the onions have softened, about 5 minutes.

Flip the mushrooms and onions over. Spoon the feta mixture into the gill side of the mushroom caps. Continue to grill until the mushrooms are completely tender and the cheese is warm, about 5 more minutes. During the last minute of grilling, place the rolls on the edge of the grill to toast until golden.

Spread the basil mayo on the toasted rolls. Build each burger with a layer of arugula, a grilled onion slice, and a stuffed mushroom cap. Serve immediately while the mushrooms are juicy and the cheese is soft.