1. 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.
2. With a fork, mix in 3/4 cup shredded cheese. 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.
3. Pat dough into an even round 3/4 inch thick. Using a 2 1/2-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.
4. Bake in a 425° oven until biscuits are puffy and golden brown, about 15 minutes. Transfer to wire racks to cool completely.
5. Cut biscuits in half horizontally. Spread cut sides with honey mustard (see notes) to taste. Fold a slice of ham on the bottom half of each biscuit, then set tops in place.