Preheat grill or grill pan to medium heat. Brush mushroom caps all over with vegetable oil and season with salt and pepper.
Place mushrooms cap side down on grill. Let cook until grill marks form, about 4 to 6 minutes, then use tongs to flip caps and cook 4 to 6 minutes more, or until mushrooms are tender.
Meanwhile, combine parsley, basil, olive oil, garlic, lemon juice and zest, and red pepper flakes in the bowl of a food processor. Season with salt and pepper and blend until sauce is smooth.
Remove mushrooms from grill and place in a large bowl. Add sauce and toss gently to coat.