Preheat oven to 350°F (175°C) and lightly spray a 9x13-inch baking dish with cooking spray.

Heat olive oil in a skillet over medium heat. Add onion and garlic; cook and stir until softened, about 5 minutes. Let cool for another 5 minutes.

Combine onion mixture, turkey, bread crumbs, egg, 1 tablespoon Worcestershire sauce, 1 ½ teaspoon parsley, allspice, and nutmeg in a bowl. Mix thoroughly and form into 1-inch meatballs. Transfer to the baking dish.

Bake meatballs in the preheated oven for 25–30 minutes, or until the internal temperature reads 165°F (74°C).

Combine beef broth and flour in a jar with a tight lid and shake to dissolve the flour.

Pour the broth mixture into a skillet over medium-high heat. Add 1 tablespoon Worcestershire sauce, parsley, black pepper, and salt. Bring to a boil, stirring frequently. Reduce to medium heat and simmer until thickened, about 2–3 minutes.