In a small bowl, combine the yeast and the warm water. Set aside and let it proof until bubbly and foamy, about 5-10 minutes.
In a large mixing bowl, whisk together the flour, sugar, baking powder, baking soda, and salt. Add the shortening and butter. Use a pastry cutter or two forks to cut the shortening and butter into the flour mixture until incorporated. The mixture should resemble coarse meal. You will still see flakes of shortening and butter throughout, which is good!
Add the yeast mixture and the buttermilk, stirring with a wooden spoon until a soft, sticky dough forms. Cover and chill for at least 2 hours, or up to 3 days.
Once the dough is chilled, turn it out onto a lightly floured surface, and roll out to about ½-inch thickness. The dough will be fairly sticky, so flour your hands, if necessary, to prevent it from sticking. Use a biscuit cutter to cut out rounds (I use a 2 -inch cutter). Firmly press the cutter down into the dough, but do not twist. Twisting the biscuit cutter seals off the edges of the biscuits and they therefore will not rise as high. Roll the scraps and continue to cut more biscuits.
Arrange the biscuits with sides touching in a 12-inch cast iron skillet, or on a baking sheet. Cover and place the pan in the fridge to chill for about 10-15 minutes while you preheat the oven.
Preheat oven to 400°F.
Bake the biscuits, uncovered, in the center of the oven until golden brown, about 15-17 minutes.
Brush the warm biscuits with melted butter. Serve!