Grilled Portobello Burgers

Using tip of paring knife, cut 1/2-inch crosshatch pattern on tops of mushroom caps, 1/16 inch deep. Combine oil, vinegar, garlic, 1 teaspoon salt, and 1/2 teaspoon pepper in 1-gallon zipper-lock bag. Add mushrooms, seal bag, turn to coat, and let sit for at least 30 minutes or up 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 (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.

FOR A GAS GRILL: Turn all burners to high, cover, and heat grill until hot, about 15 minutes. Turn all burners to medium-high.

Clean and oil cooking grate. Remove mushrooms from marinade, reserving excess. Brush onions all over with reserved mushroom marinade. Place onions and mushrooms, gill side up, on grill. Cook (covered if using gas) until mushrooms have released their liquid and are charred on first side, 4 to 6 minutes. Flip mushrooms and onions and continue to cook (covered if using gas) until mushrooms are charred on second side, 3 to 5 minutes.

Transfer onions to platter; remove toothpicks. Transfer mushrooms to platter, gill side up, and divide feta mixture evenly among caps, packing down with hand. Return mushrooms to grill, feta side up, and cook, covered, until heated through, about 3 minutes.

Return mushrooms to platter and tent with aluminum foil. Grill rolls cut sides down until lightly charred, about 1 minute. Spread basil-mayonnaise on bun bottoms and top each with 1 mushroom and 1 onion slice. Divide arugula evenly among burgers, then cap with bun tops. Serve.