Using the tip of a paring knife, score the top of each mushroom cap in a 1/2-inch crosshatch pattern about 1/16-inch deep. Combine the oil, vinegar, garlic, salt, and pepper in a large zipper-lock bag, add the mushrooms, and turn to coat. Let the mushrooms marinate at room temperature for at least 30 minutes.

Stir the feta, red peppers, and sun-dried tomatoes together in a small bowl. In a separate bowl, whisk the mayonnaise and basil. Secure each onion slice by pushing a toothpick horizontally through the rings to keep them intact while they are on the grill.

Prepare your grill for medium-high heat. If using charcoal, the grill is ready when the coals are partially covered with ash and you can hold your hand over the grate for 3 to 4 seconds. If using gas, heat the grill on high for 15 minutes, then reduce to medium-high.

Clean and oil the grill grate. Remove the mushrooms from the bag, reserving the marinade, and brush the marinade over the onion slices. Place the mushrooms, gill-side up, and the onions on the grill. Cook until the mushrooms have released their liquid and are charred on the first side, 4 to 6 minutes.

Flip the mushrooms and onions. Continue to cook until the mushrooms are charred on the second side and the onions are tender, 3 to 5 minutes. Remove the onions to a plate and pull out the toothpicks.

Transfer the mushrooms to a platter temporarily, gill-side up, and divide the feta mixture among the caps, packing it down gently. Return the mushrooms to the grill, cheese-side up, and cook covered until the cheese is heated through and the mushrooms are tender, about 3 minutes.

Spread the basil mayonnaise on the bottom of each roll. Top each with a mushroom, an onion slice, and a handful of arugula. Serve immediately while the mushrooms are still hot and juicy.