Trim any excess fat from the chicken breasts. Cut the chicken into bite-sized pieces, about 1 to 1.5 inches wide.
In a bowl, combine the whole milk and pickle juice. Add the chicken pieces and stir to coat. Cover and refrigerate for at least 30 minutes, or up to 2 hours.
In a separate bowl, whisk together the all-purpose flour, powdered sugar, salt, and black pepper.
Beat the large egg in a small bowl to prepare the egg wash.
Remove the chicken from the marinade, letting excess liquid drip off. Dip each piece into the beaten egg, then dredge in the flour mixture, pressing lightly to adhere. Repeat the egg wash and flour dredge once more for extra crispiness.
Pour vegetable oil into a deep skillet or frying pan to about 1 inch depth. Heat over medium heat until the oil reaches 350°F (175°C).
Carefully fry the coated chicken pieces in batches for about 3-4 minutes per side or until golden brown and cooked through (internal temperature should reach 165°F / 74°C).
Use tongs or a slotted spoon to remove nuggets from the oil and place on paper towels to drain excess oil. Serve warm with your favorite dipping sauces.