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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.
Spoon the batter into the muffin wells, filling each about 3/4 full. Top the muffins with the reserved ham, sprinkle with the remaining chives, and sprinkle with pepper. Top each muffin with 1/2 teaspoon of the remaining melted butter.
Bake until the tops just begin to brown and a toothpick inserted in the center of a muffin comes out clean, 20 to 25 minutes. Cool the muffins in the pan for about 5 minutes, then transfer to a cooling rack to cool completely.