Combine the flour, cornmeal, sugar, baking powder, and salt in a large bowl. In separate bowl, combine the milk, eggs, and butter. With a wooden spoon, stir the wet ingredients into the dry until most of the lumps are dissolved. Don't overmix! Mix in 2 cups of the grated Cheddar and the dill, and allow the mixture to sit at room temperature for 20 minutes.
Meanwhile, preheat the oven to 350 degrees F. Grease a 9 by 13 by 2-inch baking pan.
Pour the batter into the prepared pan, smooth the top, and sprinkle with the remaining grated Cheddar. Bake for 30 to 35 minutes, or until a toothpick comes out clean. Cool and cut into large squares. Serve warm or at room temperature.