Place the butterflied chicken breasts in a large bowl and pour in the 2 cups of pickle juice to fully submerge; cover and refrigerate for 2 to 3 hours. Drain and pat the chicken dry before breading.
Whisk the egg, milk, and hot sauce together in a medium bowl to make the wet mixture.
In another medium bowl, combine the flour, powdered sugar, salt, black pepper, paprika, and cayenne pepper; mix well to make the seasoned flour.
Heat enough peanut oil in a heavy-duty pot or deep fryer to reach 350°F (about 175°C).
Working one breast at a time, dip a butterflied chicken breast into the egg mixture, letting excess drip off, then dredge thoroughly in the seasoned flour to coat completely.
Carefully place the dredged chicken into the hot oil and fry 3 to 4 minutes per side, until golden brown and cooked through; remove and drain on paper towels. Repeat for remaining breasts, frying each immediately after dredging.
While the last chicken finishes, butter the cut sides of the brioche buns and toast them cut-side down in a skillet over medium-low heat until golden.
Assemble each sandwich: place one fried chicken breast on the bottom bun, top with 2 dill pickle chips, and add lettuce, tomato slice, and a slice of pepper jack cheese if making the deluxe version; cover with the top bun and serve.