Separate lettuce leaves carefully, keeping them whole. Rinse and pat dry. Slice tomato, thinly slice red onion, and slice avocado just before serving. Set all toppings on a platter.
Using the large holes of a box grater, grate about 2 tablespoons of yellow onion, catching the juices along with the pulp. Add both pulp and juice to your mixing bowl.
Add ground turkey, Worcestershire sauce, grated onion, minced garlic, Dijon mustard, mayonnaise, smoked paprika, salt, and black pepper to the bowl. Mix gently with your hands just until seasonings are evenly spread. Stop as soon as it comes together.
Divide mixture into 4 equal portions. Shape each into a patty about 1/2 inch (1.25 cm) thick and slightly wider than your lettuce leaves. Press a shallow dimple into the center of each patty with your thumb.
Place cast iron skillet over medium heat and add avocado oil. Let pan heat for 2 to 3 minutes until the oil shimmers.
Lay patties in the hot skillet, leaving space between them. Cook undisturbed for about 5 minutes on the first side until a browned crust forms. Flip once with a fish spatula and cook another 4 to 5 minutes on the second side. Check center with an instant-read thermometer and pull at 165°F (74°C).
If using cheese, lay a slice on each patty during the last 2 minutes of cooking. Cover skillet loosely with a lid so residual heat melts the cheese fully.
Move cooked patties to a plate and let them rest for 2 to 3 minutes. Resting lets juices settle back into the meat instead of running out when you bite in.
Lay one lettuce leaf flat, or stack two for extra sturdiness, and set a patty in the center. Top with tomato, red onion, avocado, and condiments of choice. Fold a second lettuce leaf over the top, or wrap the sides up around the patty, and serve right away.