Ingredients

• 1/2 cup chopped drained piquillo peppers or roasted red peppers from jar
• 1 tablespoon Sherry wine vinegar
• 2 teaspoons finely chopped fresh thyme
• 1 garlic clove, minced
• 1/2 cup purchased mayonnaise
• Cayenne pepper
• 6 large portobello mushrooms, stemmed, dark gills scraped out
• 1 sweet onion (such as Vidalia or Walla Walla), cut into 1/3-inch-thick slices
• Extra-virgin olive oil
• 2 teaspoons chopped fresh thyme
• 1 large garlic clove, minced
• 8 ounces Idiazábal or smoked Gouda cheese, rind trimmed, thinly sliced
• 6 sourdough rolls (each about 4 inches in diameter), halved horizontally
• 1 bunch watercress, stems trimmed

Instructions

1. Puree first 4 ingredients in mini processor. Transfer to small bowl. Whisk in mayonnaise; season with salt, black pepper, and cayenne pepper. DO AHEAD: Can be made 3 days ahead. Cover; chill.
2. For mushrooms: Arrange mushroom caps and onion slices on baking sheet; brush both sides with oil, then sprinkle with salt, pepper, and thyme. Turn mushrooms, gill side up, and sprinkle with garlic. Let stand at least 30 minutes and up to 2 hours. Prepare barbecue (medium-high heat). Place mushrooms, gill side up, and onion slices on rack. Cook until grill marks appear, about 4 minutes. Turn vegetables over, transferring to cooler part of grill. Cover and cook until mushrooms are very tender and onion slices still retain some texture, about 8 minutes. Turn mushrooms, gill side up; top with cheese. Arrange rolls, cut side down, on grill. Cover; cook until cheese on mushrooms melts and cut sides of rolls are golden, about 1 minute longer.
3. Arrange mushroom caps and onion slices on baking sheet; brush both sides with oil, then sprinkle with salt, pepper, and thyme. Turn mushrooms, gill side up, and sprinkle with garlic. Let stand at least 30 minutes and up to 2 hours.
4. Prepare barbecue (medium-high heat). Place mushrooms, gill side up, and onion slices on rack. Cook until grill marks appear, about 4 minutes. Turn vegetables over, transferring to cooler part of grill. Cover and cook until mushrooms are very tender and onion slices still retain some texture, about 8 minutes. Turn mushrooms, gill side up; top with cheese. Arrange rolls, cut side down, on grill. Cover; cook until cheese on mushrooms melts and cut sides of rolls are golden, about 1 minute longer.
5. Arrange roll bottoms on plates; spread with aioli. Top each with mushroom, onion slice, and watercress. Spread cut side of roll tops with aioli; press onto watercress.