Preheat smoker to 225°F to 250°F with hickory or oak wood. Divide ground beef into 6 equal portions (about ⅓ pound each). Gently form into patties ¾ inch thick, creating a ½-inch indentation in the center of each with your thumb. Season both sides with salt, pepper, garlic powder, and smoked paprika.
Make pimento cheese: Combine shredded cheddar, mayonnaise, pimentos, Dijon mustard, garlic powder, black pepper, and cayenne in a bowl. Mix until creamy but slightly chunky. Refrigerate 15 to 20 minutes to let flavors meld.
Make burger sauce: Whisk together mayonnaise, ketchup, mustard, pickle juice, smoked paprika, and garlic powder until smooth. Set aside.
Place seasoned burger patties on smoker grates, spacing them 1 inch apart. Close lid and smoke for 45 to 60 minutes until internal temperature reaches 125°F to 130°F for medium doneness. Check temperature starting at 40 minutes.
Preheat cast iron skillet, flat top, or grill grates to 450°F to 500°F. Transfer smoked burgers to hot surface and sear 60 to 90 seconds per side until brown crust forms. During final 30 seconds, add about ¼ cup pimento cheese to each burger and let it begin melting.
Butter brioche buns and toast cut-side down for 2 to 3 minutes until golden brown.
Assemble burgers: Spread burger sauce on bottom bun, add 3 to 4 pickle chips, place burger with melted pimento cheese on top, and finish with toasted top bun. Serve immediately while hot.