Preheat the oven to 450 degrees. Line a baking sheet with parchment paper.
In a medium-sized bowl, mix together the flour, baking powder, and salt with a fork. Using a pastry blender, cut the butter into the flour until it resembles cornmeal, with tiny bits of butter distributed throughout the dry ingredients.
Add the Cheddar cheese, Parmesan cheese, and dried chives to the flour mixture and stir to combine.
Crack the eggs into a small bowl and beat them. Remove about two tablespoons of the eggs and reserve it for later.
Add the cream to the rest of the eggs and mix it well. Add the ham to the cream and egg mixture, stirring to combine.
Make a well in the bowl of flour and pour the egg and cream mixture into the well. Using a large spoon and quickly, stirring and turning the mixture, combine the wet and dry ingredients, just until the dry ingredients are moistened and there are no pockets of dry flour. If you don’t have enough liquid, just drizzle some cream into the bowl, a little at a time.
Use your hands to press the dough into a seven or eight-inch circle on a lightly floured surface, patting the top of the dough round to make it flat.
Cut the dough into wedges if you want your scones to have the classic triangular shape. Alternately, you can cut them into squares, diamonds, or even rounds with a biscuit cutter.
Place the scones on the parchment-lined pan. Use a pastry brush to brush the reserved beaten eggs on the tops of the scones.
Bake the scones for about 12 to 15 minutes or until they are lightly browned on top. Serve warm.
Store tightly covered in the refrigerator for up to five days. Reheat by wrapping the scones in aluminum foil and placing them in a warm oven, about 250 degrees, for about 5 minutes.