Score the top of each mushroom cap with a 1/2-inch crosshatch pattern, cutting about 1/16-inch deep. Combine the oil, vinegar, garlic, salt, and pepper in a large zipper-lock bag. Add the mushrooms, seal, and turn to coat; let them marinate for at least 30 minutes at room temperature.

Stir the feta, red peppers, and sun-dried tomatoes together in a small bowl until well combined. In a separate bowl, whisk the mayonnaise and basil together. Secure each onion slice by pushing a toothpick horizontally through the rings to prevent them from falling apart on the grill.

Prepare the grill for medium-high heat. If using gas, heat with all burners on high for 15 minutes, then reduce to medium-high; if using charcoal, wait until the coals are partially covered with ash and spread them evenly over the bottom of the grill.

Clean and oil the cooking grate. Remove the mushrooms from the bag, reserving the marinade. Brush the marinade all over the onion slices. Place the onions and the mushrooms, gill-side up, on the grill and cook until the mushrooms have released their moisture and have developed dark grill marks, 4 to 6 minutes.

Flip the mushrooms and onions. Continue cooking until the mushrooms are charred on the second side and the onions are softened and marked, 3 to 5 minutes. Transfer the onions to a plate and remove the toothpicks.

Turn the mushrooms gill-side up on a platter and pack the feta mixture into the caps. Return the mushrooms to the grill, cover, and cook until the filling is heated through, about 3 minutes.

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