Adjust oven rack to middle position and preheat oven to 450 degrees. Line a baking sheet with parchment paper.
In a food processor, pulse flour, sugar, baking powder, baking soda, salt, butter, and shortening in until mixture resembles coarse meal, about 15 pulses. Transfer to large bowl. Stir in buttermilk until combined.
On a lightly floured counter, turn out dough. Knead until smooth, about 8 to 10 times. Pat dough into 9-inch circle, about 3/4 inch thick.
Using 3-inch biscuit cutter dipped in flour, cut rounds of dough (do not twist cutter) and arrange on prepared baking sheet. Gather remaining dough, pat into 3/4-inch-thick circle, and cut out remaining biscuits. (You should have 8 biscuits total.)
Bake until biscuits begin to rise, about 5 minutes, then rotate pan and reduce oven temperature to 400 degrees. Continue baking until golden brown, 12 to 15 minutes longer, then transfer to wire rack and let cool. (Biscuits can be stored in zipper-lock bag for up to 2 days.)