Heat the oven to 375°F and butter a 9-by-13-inch baking dish. Whisk the cornmeal, flour, baking powder and salt in a large bowl.
Whisk the eggs, buttermilk and melted butter in a second bowl. Pour into the dry ingredients and fold just until no dry flour remains.
Fold in the ham, 4 ounces cheddar and scallions. Spread the thick batter in the dish and scatter the remaining cheddar across the top.
Bake for 30 to 35 minutes, until the middle springs back, a tester has moist crumbs rather than wet batter and the center reaches at least 165°F.
Rest for 10 minutes before cutting into eight large squares.