In a 12-inch nonstick skillet, cook the onion and garlic in oil in over moderate heat, stirring, until the onion is softened, about 2 minutes. Add the carrot and cook, stirring, until softened, about 3 more minutes.
Add the mushrooms, ½ teaspoon salt, and ¼ teaspoon pepper and cook, stirring occasionally, until the liquid that the mushrooms give off is evaporated and they are very tender, about 10 to 15 minutes. Stir in the Worcestershire sauce, parsley, and 3 tablespoons barbecue sauce, then transfer the vegetables to a large bowl and cool.
In a small bowl, stir together the bread crumbs and milk and let it stand for 5 minutes. Stir in the egg and egg white, then add to the vegetables. Add the turkey and the remaining ½ teaspoon salt and ¼ teaspoon pepper to the vegetable mixture and mix well with your hands. (The mixture will be very moist).
Form the mixture into a 9x5-inch oval loaf in a lightly oiled 13x9x2 inch metal baking pan and brush the meatloaf evenly with the remaining 2 tablespoons of barbecue sauce. Bake in the middle of the oven until a thermometer inserted into the meatloaf registers 170°F., 50 to 55 minutes.