Combine the warm water (between 100 and 110 degrees F) and yeast with 1 teaspoon of the sugar in a bowl and set aside to proof for 5 minutes until foamy.
While the yeast is proofing, add the flour, sugar, baking powder, baking soda, and salt in a large bowl. Add the cold, cubed butter and shortening. Use a pastry cutter, two knives, or your fingers to cut the butter and shortening into the flour.
Add the proofed yeast and buttermilk and stir with a fork or spatula to combine just until the dough starts coming together. Do not overmix. The dough can be covered tightly with plastic wrap and stored in the fridge for up to 1 week at this point.
Turn the biscuit dough out onto a floured surface and sprinkle the top with more flour. Dust your hands with some flour to keep them from sticking, then pat the dough out into a large rectangle about 1-inch thick.
Cut out biscuits using a sharp 2 to 2 1/2-inch round biscuit cutter and pressing straight up and down without twisting. You may need to dip your cutter into flour before cutting the biscuits to prevent them from sticking. Cut carefully to get as many biscuits as you can out of the first cutting. Scraps can be squished together, patted out, and cut out, but they won't be quite as light and tender as the first ones.
Arrange the biscuits in the prepared skillet or on a baking sheet with the sides touching. This will help them rise even more and give nice soft sides. If you have additional biscuits that don't fit in your skillet, you can always refrigerate and bake them later or bake them separately on a baking sheet.
Bake for 14-16 minutes until golden brown on top and cooked through. Remove from the oven and brush with an additional 3-4 tablespoons of melted butter, if desired.