Preheat oven to 400°F. Line a sheet pan with parchment paper.

In a large bowl, combine the flours, baking powder, baking soda, and salt. Add the butter and toss in the flour mixture until coated. Use a pastry blender or knife to blend the butter into the flour, until the butter is the size of small peas. Add the cheese. Pour in the buttermilk and use a wooden spoon and then your hands to bring the mixture together into a ball. Do not overwork the dough, it's alright if there are some dry bits.

Turn dough out onto a clean surface and pat into a disk about 1 inch thick. Use a round cutter to cut out biscuits, or shape in a rectangle and cut into squares with a knife. Place on prepared sheet pan and brush with buttermilk.

Bake until golden brown, about 20 minutes. Serve warm or at room temperature. Store in an air tight container. Biscuits are best the day they are baked but will keep for a few days.