Slice each chicken breast in half horizontally to create 4 thinner cutlets about 1/2 inch thick. Pound any uneven pieces between plastic wrap until uniform thickness.
Place chicken cutlets in a zip-top bag or bowl and pour pickle juice over them, ensuring all pieces are submerged. Cover and refrigerate for at least 1 hour, up to 4 hours.
In a large shallow bowl, whisk together flour, powdered sugar, cayenne, paprika, chili powder, garlic powder, onion powder, mustard powder, salt, black pepper, and baking powder until thoroughly combined.
In a separate shallow bowl, whisk together the egg, milk, and hot sauce until smooth and fully combined.
Remove chicken from marinade, drain off pickle juice, and pat each cutlet thoroughly dry with paper towels on both sides and all edges.
Working one piece at a time, dip chicken into egg wash, letting excess drip off. Transfer to flour mixture and press firmly, coating all surfaces. For extra crunch, repeat the egg wash and flour dredge one more time.
Pour peanut oil into a Dutch oven to a depth of 2 inches and heat to 350°F (175°C), using a deep-fry thermometer to monitor the temperature.
Carefully lower one or two breaded cutlets into the hot oil, frying for 3-4 minutes per side until deep golden-brown and internal temperature reaches 165°F (74°C). Drain on a wire rack.
Melt butter in a skillet over medium heat and toast the brioche buns cut-side down for 1-2 minutes until golden and slightly crispy.
Place one fried chicken cutlet on each toasted bottom bun, top with 2-3 dill pickle chips, and cover with the top bun. Serve immediately while hot and crispy.