Cut the chicken breasts in half horizontally to make 4 thin cutlets. Place the cutlets between 2 pieces of heavy-duty plastic wrap and pound to 1/8-inch thick with a mallet or heavy skillet. Season the chicken with 1 teaspoon salt and 1/2 teaspoon each pepper and paprika.
Heat about 2 inches peanut oil in a heavy-bottomed pot over medium heat until a deep-fry thermometer registers 325 degrees F. Meanwhile, whisk the egg, milk and 2 tablespoons water in a baking dish. Whisk both flours, the malted milk powder, confectioners' sugar, baking soda, dry mustard, the remaining 1/2 teaspoon paprika and 1 teaspoon each salt and pepper in another dish. Toss the pickle slices, pickle juice and vinegar in a bowl and set aside.
Working in batches, dip the chicken in the egg mixture, turning to coat, then dredge in the flour mixture and shake off any excess. Fry the chicken in the hot oil until golden brown, about 4 minutes, adjusting the heat as needed. Drain on paper towels.
Heat a large skillet over medium heat. Spread the cut sides of the buns with some of the butter and lightly toast in the skillet, buttered-side down; spread with more butter. Put 2 pickle slices on each bun bottom; top with a piece of chicken and cover with the bun tops.