Heat oven to 375°F; butter 9-by-13-inch dish. Whisk cornmeal, flour, baking powder, and salt in large bowl.

Separately whisk eggs, buttermilk, and melted butter; fold into dry ingredients just until no dry flour remains.

Fold in ham, 4 ounces cheddar, and scallions. Spread thick batter in dish; scatter remaining cheddar over.

Bake 30 to 35 minutes until middle springs back, tester shows moist crumbs without wet batter, and center at least 165°F.

Rest 10 minutes; cut eight large squares.