For the chicken: Whisk together the milk, salt, hot sauce, pepper and eggs in a large mixing bowl. Add the chicken breasts and turn to coat. Cover the bowl with plastic wrap and refrigerate for at least 1 hour or up to 24 hours.
For the biscuits: Position an oven rack in the middle of the oven and preheat to 425 degrees F. Brush a metal 8-inch square baking pan with 1 tablespoon of the melted butter. Whisk the flour in a large mixing bowl. Work the shortening into the flour with your fingers or a dough cutter until there are no large lumps. Gently stir the buttermilk into the flour mixture until it comes together. Knead the dough in the bowl, turning 5 to 7 times. Evenly and gently press the dough into the prepared pan. Brush the top with 1 tablespoon melted butter. Using a bench scraper or butter knife, cut the dough evenly into 4 squares. Bake, brushing with the remaining 1 tablespoon of melted butter halfway through, until golden brown on top and cooked through, 20 to 25 minutes. Cool for 5 minutes on a rack. Invert the biscuits onto the rack, then cover loosely with aluminum foil to keep warm while you fry the chicken. Lower the oven to 325 degrees F.
Finish the chicken: Fill a large cast-iron pan halfway up with oil and heat to 350 degrees F on medium-high heat. Place the flour, paprika, granulated garlic and salt in a large resealable plastic bag. Remove the chicken from the egg mixture, shaking off any excess. Place in the bag and seal. Shake until the chicken is evenly coated. Carefully place the chicken in the hot oil and fry, turning halfway through with a pair of tongs until an instant-read thermometer inserted into the thickest part of each chicken breast registers 155 degrees F, 18 to 20 minutes. Transfer the chicken to a paper-towel-lined plate to drain. Season generously with salt. Cool for 5 minutes.
To assemble, cut the biscuits into 4 pieces, then halve each horizontally and put the bottoms on a baking sheet. Use a pair of tongs or a knife to remove the bone from each of the chicken breasts. Use your hands to tear the chicken breasts into 3 long pieces, keeping the skin as intact as possible. Place the pieces from one chicken breast on each of the biscuit bottoms and top each with a slice of cheese. Heat them in the oven until the cheese melts, 3 to 5 minutes. Cover each with the biscuit tops. Serve immediately.
Inspired by the Chicken Cheddar Biscuit from Time-Out for Cooking Channel