Position a rack in the center of the oven and heat to 400°F.

Warm the olive oil in a large nonstick skillet over medium heat. Add the onion and garlic and cook, stirring often, until the onion is soft and lightly golden, about 2 minutes.

Stir in the carrot and continue cooking for about 3 minutes, until the carrot begins to soften.

Add the mushrooms, 1/2 teaspoon of the salt, and 1/4 teaspoon of the pepper. Cook, stirring occasionally, until the mushrooms have released their liquid and are very tender, 10 to 15 minutes.

Take the skillet off the heat and stir in the Worcestershire sauce, parsley, and 3 tablespoons of the barbecue sauce. Scrape the mixture into a large bowl and let it cool for a few minutes.

In a small bowl, combine the bread crumbs and milk; let stand for 5 minutes until the crumbs absorb the milk.

Whisk the egg and egg white into the soaked bread crumb mixture.

Add the ground turkey, the remaining 1/2 teaspoon salt and 1/4 teaspoon pepper, and the bread crumb mixture to the cooled vegetables. Use your hands to mix gently until everything is evenly combined; the mixture will be quite moist.

Shape the meatloaf into a 9x5-inch oval and place it in a lightly oiled 13x9x2-inch metal baking pan. Brush the remaining 2 tablespoons of barbecue sauce over the top.

Bake on the center rack until an instant-read thermometer inserted into the center reaches 170°F, about 50 to 55 minutes.

Let the meatloaf rest for 5 minutes before slicing so the juices can settle.