Preheat the oven to 350°F. Prepare a 15 inch cast iron skillet by spraying with cooking spray or coating with butter. (See notes for instructions on baking in other pans.) Set aside.
In a medium saucepan, melt butter over medium heat. Once completely melted, reduce the heat slightly and let it simmer until it turns golden-brown and smells nutty and fragrant. Set aside and allow to cool slightly.
Meanwhile, in a large bowl, whisk together the all-purpose flour, sugar, baking powder, baking soda, sea salt, and cornmeal until well combined.
To the cooled butter, add the room temperature buttermilk and eggs and whisk until well combined. Fold in the shredded zucchini that has had the excess liquid squeezed out.
Add the wet ingredients to the dry ingredients and fold together just until no dry spots remain, being careful not to overmix. The batter will be very thick.
Scoop the batter into the prepared cast iron skillet and spread it evenly. Bake for 23 to 25 minutes, or until a skewer inserted in the center comes out mostly clean. (It's a moist bread, so some sticky crumbs are fine—just make sure there's no uncooked batter.)
Remove from oven and cool in the pan for about 15 minutes before removing and slicing.