Preheat oven to 375°. Cut butter into cubes, then brown the butter in a cast-iron skillet over medium heat. Swirl butter all over the sides and corners of pan. Pour butter into a large bowl. Place the skillet in the oven to heat before baking.
Whisk syrup into the brown butter. Whisk buttermilk into the mixture (which should be cool to the touch at this point). Whisk in eggs until smooth.
In a medium bowl, combine cornmeal, flour, baking powder, salt, and baking soda. Whisk into wet ingredients until just incorporated.
Remove skillet from oven. Pour batter into skillet and bake until a toothpick comes out clean, 25 to 30 minutes. Serve with Jalapeño Butter.