Heat oven to 425° F. Butter a 9-by-13-inch baking pan.
In a medium bowl, whisk together the flour, cornmeal, baking powder, baking soda, and salt. Make a well in the center of the flour mixture and add the milk, eggs, ½ cup of the maple syrup, and ½ cup of the butter; whisk together the wet ingredients, then incorporate the dry ingredients until just combined (do not overmix).
Transfer the batter to the prepared pan and bake until a toothpick inserted in the center comes out clean, 20 to 25 minutes.
Meanwhile, in a small bowl, combine the remaining ¼ cup of butter and ¼ cup of maple syrup. After removing the corn bread from oven, brush with the butter mixture. Cool completely in the pan, then cut into pieces.