Preheat the oven to 400 degrees F. Grease a 9 by 9-inch baking dish with butter.
In a medium bowl, whisk together the flour, cornmeal, sugar, baking powder, and salt. In another medium bowl, whisk together very well the eggs, milk, butter, applesauce, and maple syrup. Pour the wet ingredients into the dry ingredients and stir to incorporate (do not overmix). Pour into the prepared dish and bake until a toothpick inserted into the center comes out clean, about 25 minutes. Serve with Maple Lime Butter.
Maple Lime Butter:
In a small bowl, mix the ingredients together until well incorporated. Refrigerate for at least 10 minutes to firm up slightly before serving.