Copycat Chick-fil-A Chicken Sandwich

In shallow baking dish, whisk egg, milk, ½ teaspoon paprika, and water together. In another baking dish, whisk both flours, powdered milk, powdered sugar, baking soda, dry mustard, and remaining ½ teaspoon paprika and season with salt and pepper. Meanwhile, heat about 2" of peanut oil to 325° in a heavy-bottomed pot or cast iron skillet. While oil heats, slice pickle, and set aside. Working in batches, dip chicken in the egg mixture, turning to coat, then dredge in flour mixture and shake off any excess. Fry the chicken in hot oil, using a candy thermometer to monitor oil temperature, until golden brown, about 4 minutes. Drain on paper towels. Heat a large skillet over medium heat. Spread the cut side of the buns with some butter, and lightly toast in the skillet, buttered-side down. To assemble sandwiches, spread grilled buns with more butter, dip 2 pickle slices in jarred pickle juice to moisten and place on the bottom bun. Top with a piece of fried chicken and the bun top.