Arrange a rack in the middle of the oven and heat to 375°F. Line a standard 12-well muffin pan with papers liners or coat the wells with cooking spray.
Whisk together the flour, cornmeal, baking powder, garlic powder, baking soda, and salt in a large bowl. Whisk the buttermilk, eggs, and 5 tablespoons of the butter in a medium bowl until combined. Pour the wet ingredients into the dry, and mix with a wooden spoon or rubber spatula until just combined. Some lumps are fine. Fold in the cheese, all but 1/2 cup of the ham, and 2 tablespoons of the chives.