Begin by marinating the chicken. Place the 4 chicken pieces in a ziplock bag and pour in 1 cup of pickle juice. Ensure the chicken is well-coated, seal the bag, and refrigerate for at least 30 minutes.
Combine 1 cup all-purpose flour, 3 tablespoons powdered sugar, 1 teaspoon paprika, 1 teaspoon salt, 1-2 teaspoons cayenne pepper, 1 teaspoon baking powder, and freshly ground black pepper in a bowl to make the breading mixture.
In another bowl, whisk together 1 large egg and 1 cup milk to create an egg wash.
Heat 2-3 cups of vegetable oil in a large skillet or deep fryer to 350 degrees for deep frying.
Remove the chicken from the marinade, and gently shake off any excess pickle juice. Dredge each piece of chicken in the dry mixture, then dip into the egg mixture, and coat again in the flour mixture.
Carefully lower the chicken into the hot oil and pan fry for 3-4 minutes on each side or until the coating is golden and the chicken is cooked through.
While the chicken is frying, toast the 4 hamburger buns.
For the homemade Chick-fil-A sauce, combine 1/2 cup mayonnaise, 2 tablespoons barbecue sauce, 2 tablespoons honey, 1 tablespoon yellow mustard, and 1 teaspoon lemon juice in a bowl and whisk until smooth.
Once the chicken is done, let it rest on a wire rack to maintain crispiness. Assemble the sandwich by placing the fried chicken on the bottom bun, topping it with two dill pickle slices, and spreading the Chick-fil-A sauce on the top bun before serving.