Whisk together cornmeal, flour, sugar, baking powder, salt, and baking soda in a large bowl. Add Cheddar and scallions and whisk until combined.
Whisk together whole milk, buttermilk, eggs, and butter in another bowl, then add to cornmeal mixture and stir until just combined (do not overmix).
Pour batter into a well-greased 9- by 2-inch round cake pan and bake in middle of oven until pale golden and a tester inserted in center comes out clean, 18 to 20 minutes.
Cool corn bread in pan on a rack 10 minutes, then turn out onto rack and invert onto a plate before cutting into wedges.
