Preheat oven to Low Broil.

Heat oil in a large, deep oven–proof skillet over medium-high heat. Add onion and cook, stirring constantly, 1 minute. Add ground turkey and cook, crumbling with a spoon until lightly browned, about 7 minutes.

Season with salt, 1/2 teaspoon pepper, garlic powder, 3 tablespoons ketchup, and Worcestershire sauce and stir to combine. Add broth and water and bring to a boil, stirring often to loosen any browned bits from the bottom of the pan.

Stir in rice and return to a boil. Cover and reduce heat to medium-low and simmer until liquid is absorbed and rice is tender, about 20 minutes. Add up to 1/4 cup additional water if needed if liquid is evaporated before rice is tender.

Just before it is finished, stir in 1 1/4 cups of cheese to rice mixture and top with remaining cheese.

Broil in the preheated oven just until cheese is melted and bubbly, about 3 minutes. Let stand for 5 minutes before serving.

Meanwhile, for burger sauce, add remaining pepper and remaining ketchup with mayonnaise and mustard in a small bowl. Stir until well combined and set aside. Serve topped with pickles, lettuce, and burger sauce.