Pound cutlets evenly to 1/2-inch thickness; refrigerate in pickle brine 1 hour.

Heat peanut oil in a deep skillet to 350°F. Whisk milk and egg in one bowl; mix flour, sugar, paprika, and salt in another.

Drain chicken, dip in milk mixture, and press firmly into flour until craggy bits cling.

Fry cutlets one at a time for 4 to 6 minutes, turning once, until deeply golden and centers reach 165°F.

Drain on a rack 3 minutes; serve each on a bun with 3 pickle chips.