Preheat the oven to 425 degrees.
In a large bowl, whisk together flour, baking powder, baking soda, salt, and sugar.
Using the back of a fork or a pastry cutter, cut the cold butter and shortening into the dry ingredients until you have flour-covered, pea sized pieces. The mixture should look fairly crumbly at this point.
Add buttermilk and beaten egg to the bowl and stir gently until just combined. Do not over mix. Dough should be wet and sticky.
Dust a clean, dry cutting board or your kitchen counter with flour. Turn the dough out onto the floured surface. Sprinkle a bit more flour on to the top of the dough and press the dough into a rectangle. Using a dough scraper, cut the dough into 4 equal sized squares, and stack the squares on top of each other to form layers. Sprinkle a bit more flour on top of the dough if needed, then gently press it out again.
Fold the dough in half, and gently press it out again. Dust with a bit more flour if needed so that your hands don't stick to the dough. Fold in half again, and press it out again.
Gently shape the dough into a circle or rectangle shape about 3/4" thick. Use a biscuit cutter or a round cookie cutter to cut the dough into individual biscuits. Add the biscuits to a 12" cast iron skillet so that they are barely touching each other.
Bake the biscuits at 425 degrees for about 12-15 minutes, or until the tops are lightly golden brown.
Remove the skillet from the oven and immediately brush the tops of the biscuits with melted butter. Keep biscuits warm until ready to serve.