Preheat the oven to 400° F. Line a baking sheet with parchment paper. Place flour, baking powder, baking soda, and salt in the bowl of a food processor and pulse to combine. Add the diced butter and pulse until the mixture forms small pebbles.
Turn the flour mixture into a bowl and add the dill and cheese. Mix in 1 cup buttermilk, then mix in the remaining buttermilk a little at a time.
Turn dough out onto another piece of parchment paper, sprinkle a little flour on top to keep your hands from sticking to the dough, and lightly press out with your hands until about 1 inch high. Using a biscuit cutter, cut into rounds. Gather and reuse the scraps. Place the biscuits on the prepared baking sheet and bake until the tops are lightly golden and a toothpick inserted in the center of a biscuit comes out clean, 16 to 20 minutes.