Heat oven to 375 degrees.
On the stovetop, in a 11- or 12-inch skillet (ovenproof and preferably cast iron), melt the butter over medium heat.
Cook, swirling pan to lightly coat sides and bottom, until the foam subsides and the butter turns a deep nut brown. (Watch carefully to see that it does not burn.)
Pour brown butter into a large bowl. (Do not wipe out the pan.)
Whisk the maple syrup into the butter, then whisk in buttermilk.
The mixture should be cool to the touch; if not, let cool before whisking in the eggs.
Then whisk in the cornmeal, flours, baking powder, salt and baking soda.
If the skillet is no longer hot (cast iron retains heat longer than other metals), reheat it briefly on the stove for a few minutes.
Scrape batter back into it.
Bake until the top is golden brown and a toothpick inserted into it emerges clean, 30 to 40 minutes.
Cool in the skillet for 10 minutes before slicing.