Position an oven rack in the center of the oven and preheat to 425 degrees F.
Pulse together the flour, baking powder, sugar and 1 1/2 teaspoons each salt and pepper in a food processor. Add the butter and pulse until pea-sized pieces form. Add the cheese and chives and pulse until just combined. Add the buttermilk and pulse a couple of times until the dough just comes together but is not fully incorporated.
Turn the dough out onto a lightly floured surface and pat together gently into a ball. Use your hands to divide the dough into 6 even pieces. Generously coat a large baking sheet with cooking spray.
Using the baking sheet as your work surface, pat each piece of dough into a 3 1/2-inch round about 3/4 inch thick. Arrange 3 dough rounds along the top long edge of the baking sheet, spacing them out evenly and making sure there is a 1/2-inch space between the dough rounds and the edges of the baking sheet. Arrange the remaining 3 dough rounds in the same fashion along the bottom long edge of the baking sheet. Cut the center out of each round with a 2-inch round cookie cutter. Arrange these smaller rounds evenly across the middle of the baking sheet.
Brush the tops of all the dough pieces with buttermilk and bake until the smaller biscuits are golden, 12 to 15 minutes. Remove them and transfer to a plate.
Lay a piece of ham over each of the large rounds. Push each ham piece down into the hole so that it forms a cup. Crack 1 egg into each of the ham cups. If a little of the egg white spills over, it's okay. Sprinkle the eggs with salt and pepper. Bake, rotating the pan front to back after 5 minutes, until the whites are just set and the yolks are still jiggly when you lightly shake the baking sheet, 8 to 12 minutes more.
Use a spatula to remove them to individual plates and top each with a smaller biscuit piece for dipping into the egg.