Position a rack in the middle of the oven and heat the oven to 375 degrees F. Line a baking sheet with parchment.

In a large bowl, whisk the flour, baking powder, baking soda, and salt until well blended. With a pastry blender or two table knives, cut the butter into the flour mixture until its texture is fine and looks a little like cornmeal, with some butter chunks the size of peas. Stir in the chives and grated cheese. Add the buttermilk and stir with a wooden spoon until the dough is wet, sticky, and shaggy; small areas of dry ingredients are fine. Turn the dough out on to an unfloured board and knead about 12 times until the dough comes together and is smooth. Lightly flour the board and pat the dough into a 1-inch-thick square. Use a sharp knife to trim the edges of the square; this helps the biscuits rise. Cut the square into nine smaller squares. Transfer the biscuits to the lined baking sheet. Bake until firm and lightly golden brown on top and golden brown on bottom, 18 to 23 minutes.