Adjust oven rack to lower-middle position. Preheat oven to 450 degrees.
Whisk flour, baking powder, baking soda, and salt in large bowl. Add shortening to flour mixture; break up chunks with fingertips until only small, pea-sized pieces remain. Working in batches, drop butter slices into flour mixture and toss to coat; pick up each slice of butter and press between floured fingertips into flat, nickel-sized pieces. Repeat until all butter is incorporated; toss to combine. Freeze mixture (in bowl) until chilled, about 15 minutes.
Spray work surface with nonstick cooking spray and spread spray evenly across surface with kitchen towel or paper towel. Sprinkle 1/3 cup of extra flour across sprayed surface and gently spread to form an even coat.
Add all but 2 tablespoons of buttermilk to flour mixture; stir briskly with fork until ball forms and no dry bits of flour are visible, adding remaining buttermilk as needed (dough will be sticky and shaggy but should clear sides of bowl). With rubber spatula, transfer dough onto center of prepared work surface, dust surface lightly with flour, and, with floured hands, bring dough together into cohesive ball (BUT DO NOT KNEAD).
Pat dough into approximate 10-inch square; roll into 18 by 14-inch rectangle about 1/4 inch thick, dusting dough and rolling pin with flour as needed. Using a bench scraper or thin metal spatula, fold dough into thirds, brushing any excess flour from surface; lift short end of dough and fold in thirds again to form approximate 6 by 4-inch rectangle. Rotate dough 90 degrees, dusting work surface underneath with flour; roll and fold dough again, dusting with flour as needed.
Roll dough into 10-inch square about 1/2 inch thick; flip dough and cut nine 3-inch rounds with floured biscuit cutter, dipping cutter back into flour after each cut (DO NOT TWIST CUTTER WHEN MAKING THE ROUNDS). Carefully invert and transfer rounds to ungreased baking sheet, spaced 1 inch apart. Gather dough scraps into ball; roll and fold once or twice until scraps form smooth dough. Roll dough into 1/2-inch-thick round; cut three more 3-inch rounds and transfer to baking sheet. Discard excess dough.
Brush biscuit tops with melted butter. Bake, without opening oven door, until tops are golden brown and crisp, 15 to 17 minutes. Let cool on baking sheet 5 to 10 minutes before serving.