Prepare the chicken: Separate any legs into thighs and drumsticks; cut breasts crosswise in half.
In a medium bowl, whisk milk or butter milk, eggs, and salt.
Mix flour, salt, and black pepper together on a plate.
Toss the chicken pieces in the flour mixture, then coat with the egg mixture. Lift the pieces out of the egg, letting the excess drip off. Coat with the flour mixture again and transfer the pieces to dry on a rack set over a baking heat at room temperature for 30 minutes or, for a crispier result, refrigerate overnight (refrigerate the egg mixture along with the chicken; bring both chicken and egg mixture to room temperature before frying.)
Preheat the oven to 200°F. Heat in a deep-fryer or deep heavy pot over medium-high heat to 350°F.
Place in the hot fat and fry until well browned and the internal temperature reaches 160°F, turning the pieces several times with tongs keeping the fat between 320° and 360°F. Transfer the pieces to a rack set over a baking sheet lined with paper towels or a brown paper bag, and hold in the warm oven. Repeat the double-dredging procedure with the thighs and the drumsticks, frying the pieces until browned and the internal temperature reaches 175°F.