Preheat the oven to 400 degrees . Using a food processor, grind one of the toasted rolls into crumbs. Pour into a large bowl. Add the carrot, onion and garlic to the food processor and grind until chopped.
In a large, ovenproof skillet, heat the olive oil over medium heat. Add the chopped vegetables and cook until softened, about 5 minutes. Combine with the bread crumbs.
Stir the turkey, 2 tablespoons ketchup, 1 tablespoon Worcestershire sauce, the egg and the salt into the vegetable mixture. Spread the mixture in the skillet and bake until the meat is cooked through, 20 to 25 minutes. Let cool slightly.
In a small bowl, combine the remaining 2 tablespoons ketchup and 1 tablespoon Worcestershire sauce. Spread on the cut sides of the rolls.
Cut the meatloaf into long strips. Layer the bun bottoms with the lettuce and meatloaf; cover with the bun tops.