Mix the butter, Cajun seasoning, Worcestershire, and ground turkey in a bowl. Shape into four even balls. Refrigerate for at least 15 minutes.
Make the burger sauce. Mix the mayo, ketchup, garlic powder, Worcestershire, smoked paprika, brown sugar, apple cider vinegar, and salt together in a small bowl. Set aside.
Prepare your parchment paper. Cut two "burger-sized" ~5 inch x 5 inch squares of parchment paper and set aside.
Take the burgers out of the refrigerator and heat a cast iron skillet or non-stick skillet on medium-high heat.
When the skillet is hot, place two burgers on the skillet, evenly spaced apart. Sprinkle with salt, then lay your piece of parchment paper over top one of the burgers, and press down with a cast iron press, smaller cast iron skillet, or spatula to “smash” the burgers to about a half inch thick. Remove the skillet and parchment paper. Repeat with the other burger.
Once the burgers have browned and become crispy on the bottom, about two minutes, flip them and place a slice of cheese on top of each one. Let cook for another two minutes, or until crispy and browned.
Repeat this process with the last two burgers.
Assemble the burgers on your brioche buns and top with tomato, lettuce, and burger sauce.