In a bowl, mix flour, baking powder, baking soda, and salt. With two knives, a pastry blender, or your fingers, cut or rub in butter until mixture resembles coarse meal.
With a fork, mix in parsley, chives, and tarragon. Pour in buttermilk and stir gently until mixture comes together. Scrape onto a lightly floured surface and, with lightly floured hands, knead gently just until dough forms a ball.
Pat dough into a 1/2-inch-thick round. Using a 3-inch round cutter or wineglass, cut out biscuits. Gather scraps, pat out again, and cut out remaining biscuits; you should have 8 total. Space evenly on a buttered 10- by 15-inch baking sheet and sprinkle evenly with remaining shredded cheese.
Bake in a 425° oven until biscuits are puffy and golden brown, about 15 minutes. Transfer to wire racks to cool completely.