Preheat the oven to 425°. Line a 13'' x 18'' baking sheet with a baking sheet with parchment paper; set aside. Pulse flour, baking powder, salt, and sugar in a food processor to combine. Add the chilled butter pieces and pulse until the mixture resembles coarse crumbs. Transfer to a large stainless steel mixing bowl and add the chopped scallions, ham, and cheese; toss to combine.
In a small bowl, whisk the egg and buttermilk to combine and drizzle buttermilk over the top; using a fork or a rubber spatula, stir until it forms a moist, slightly tacky dough; if the dough seems dry, add a few more tablespoons of buttermilk. Do not overwork! (Alternatively, cut the butter into the flour in a large mixing bowl using a pastry cutter or two forks).
Transfer the dough out onto a lightly floured surface, dust the top of the dough with a bit more flour and bring it together gently into a rough ball. Pat the dough into a rectangle about ¾'' thick. Then, using a sharp knife or bench scraper, cut the dough into 3 pieces pieces.
Stack dough pieces on top of one another, sandwiching any loose dry bits of dough between layers, and press down to flatten. Lightly flour the top of the dough, then lightly pat or roll out the dough into a ¾" thick rectangle ( do not overwork it, or the biscuits will be tough).
Dust a 2-½-inch round biscuit cutter with flour and cut out rounds. Alternatively, use a knife to cut the biscuits into 8 to 12 rough squares. Re-stack any scraps, re-roll, and cut out more biscuits. Repeat this process until the dough is gone.
Lightly brush them with buttermilk and bake for about 15 minutes or until the biscuits are lightly golden on top and a golden brown on the bottom. Remove the Savory Biscuits from the oven, and brush with melted butter, if desired. Enjoy warm!