Start to finish: 50 minutes
4 portobello mushroom caps, gills removed
1/2 cup extra-virgin olive oil
3 tablespoons red wine vinegar
1 garlic clove, minced
1 teaspoon salt
1/2 teaspoon pepper
4 ounces feta cheese, crumbled
1/2 cup jarred roasted red peppers, patted dry and chopped
1/2 cup oil-packed sun-dried tomatoes, patted dry and chopped
1/2 cup mayonnaise
1/2 cup chopped fresh basil
4 slices red onion
4 kaiser rolls, split and toasted
1 ounce baby arugula
Cut 1/16 inch-deep slits on top of mushroom caps, spaced ½ inch apart, in crosshatch pattern. Combine mushrooms, oil, vinegar, garlic, salt, and pepper in 1 gallon zipper-lock bag, seal bag, and turn to coat. Let sit for 30 minutes to 1 hour.
Combine feta, red peppers, and sun-dried tomatoes in bowl. Whisk mayonnaise and basil together in separate bowl. Push 1 toothpick horizontally through each onion slice to keep rings intact while grilling.
For a charcoal grill: Open bottom vent completely. Light large chimney starter filled with charcoal briquettes. 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.
For a gas grill: Turn all burners to high, cover, and heat grill until hot. Turn all burners to medium-high.
Clean and oil cooking grate. Remove mushrooms from marinade, and brush onions all over with remaining mushroom marinade. Place onions and mushrooms, gill side up, on grill. Cook until mushrooms have released their liquid and are charred on first side. Flip mushrooms and onions and continue to cook until mushrooms are charred on second side.
Transfer onions to platter and discard toothpicks. Transfer mushrooms to platter, gill side up, and divide feta mixture evenly among caps, packing down mixture. Return mushrooms to grill, feta side up, and cook, covered, until heated through.
Return mushrooms to platter and tent with aluminum foil. Spread basil mayonnaise evenly over roll bottoms and top each with 1 mushroom and 1 onion slice. Divide arugula evenly among burgers, then cap with roll tops. Serve.