Preheat oven to 450 degrees. In a food processor, pulse flour, baking powder, baking soda, and salt until combined. Add cheese, butter, and chives; pulse until mixture resembles coarse meal, with a few pea-size pieces of butter remaining. Add buttermilk and pulse until combined (do not overmix).
Turn dough out onto a lightly floured work surface and knead until it just comes together. With a floured rolling pin, roll out dough 3/4 inch thick. With a floured 2 3/4-inch biscuit cutter, cut out 16 rounds (reroll and cut scraps). Place biscuits, 1 1/2 inches apart, on a baking sheet and bake until puffed and golden, 13 to 15 minutes. Transfer to a wire rack and let cool slightly. Serve warm or at room temperature.
Mix It Up
Substitute Parmesan and chopped fresh sage for the cheddar and chives.