1For carrot oil:
Preheat oven to 350F. Stir carrots and oil together in a small loaf pan. Season with salt and pepper. Roast for 45 minutes, or until carrots are very soft. Let cool to room temperature. Puree in blender and adjust seasonings to taste.
2For the stuffing:
In a large bowl, whisk together the mascarpone, egg white, lemon juice, Worcestershire, Old Bay, salt and hot sauce. Stir in the green onions, bell pepper and parsley.
3Stir in the crab and Panko; taste and adjust seasoning. Cover and refrigerate for at least 1 hour.
4Preheat broiler. Fill each mushroom cap with the crab mixture and place on broiler pan or baking sheet. Broil for 10 to 12 minutes, until browned and heated through.
5Transfer to a platter and drizzle top of each mushroom with a little carrot oil mixture and a pinch of minced parsley. Serve hot.