Sift the flour, then scoop and level. Place the flour, sugar, baking powder, baking soda, and salt in a food processor. Pulse to mix.
Next, add the cold shortening and 1/4 cup cold butter. Pulse again until the fat is mixed into the flour creating a pebble texture. Continue pulsing as you add in the cold buttermilk.
Once the dough is thoroughly mixed, stop pulsing. The dough should look wet, but even.
Dump the dough out onto a floured work surface. Press it flat with your hands until it is smooth, in an even 2-inch thick disk. Use a drinking glass, or 3-inch cutter, to cut out 8 biscuits. If needed, cut out 7, then gather the dough and cut out the last biscuit.
Place the biscuits in the skillet. Then brush the tops with the melted butter, and place in the oven immediately.
Bake in the center of the oven until the tops are golden and the biscuits have risen at least 1 inch above the top of the skillet. Around 30-35 minutes. Make sure to move the oven racks, if need, to allow room for growth.
Use a spatula or knife to separate the biscuits before lifting out of the pan.
Biscuits have the best texture when: A. the dough is cold as it goes in the oven, B. hasn't been over mixed, and C. the flour is well-sifted. Use cold ingredients and work fast! Do not over-mix the dough in the food processor or when cutting. If possible, cut all the biscuits out without re-gathering the dough. To ensure the dough is cold, you can prep the biscuits then place in the refrigerator to chill before baking.