Combine the mayonnaise, ketchup, yellow mustard, dill pickle juice, and garlic powder in a small bowl. Stir until smooth, taste, and adjust seasoning. Cover and refrigerate while you prepare the patties.
Add the ground turkey to a large mixing bowl with minced garlic, Worcestershire sauce, onion powder, smoked paprika, cumin, salt, and black pepper. Mix gently with clean hands or a fork until just combined. Divide into 4 equal portions and shape into 1/2-inch thick patties, pressing a slight indent into the center of each.
Heat 1 tablespoon of olive oil in a large cast iron skillet or non-stick pan over medium-high heat until shimmering. Carefully place the patties in the pan and cook for 5-6 minutes on the first side without moving. Flip and cook for another 4-5 minutes. If adding cheese, place a slice on each patty during the last 1-2 minutes and cover the pan to melt. Ensure internal temperature reaches 165°F (74°C). Let rest for 2-3 minutes.
Gently separate the butter lettuce leaves, choosing the largest cup-shaped ones. Wash and dry the leaves completely. Slice the tomato, avocado, and red onion and arrange on a plate.
Lay two butter lettuce leaves on a plate or cutting board, stacking them together. Spread about 1 tablespoon of the burger sauce inside the lettuce cup. Place one turkey patty inside, then top with tomato slices, avocado slices, red onion, and dill pickle rounds. Fold the edges of the lettuce up around the patty and serve immediately.