Adjust an oven rack to the lower-middle position. Place a 10-inch cast iron skillet on this rack, then crank the oven to 450°F.
Spin the grits in the carafe of a blender on high for 30 seconds. Transfer to a large bowl and add the cornmeal, salt, baking powder, and baking soda. Set aside.
In a separate bowl or large measuring cup, thoroughly combine the buttermilk and eggs. Add the wet mixture to the dry mixture and whisk smooth.
Open the oven and pull out the skillet. Swirl the butter around the bottom and sides of the skillet to both melt the butter and grease the skillet, then dump the butter into the bowl of batter. Stir to combine, then quickly pour the batter into the pan. Jiggle the skillet to evenly distribute the batter, then place back on the lower middle rack of the oven.
Bake until the cornbread is golden brown and springs back when touched, 25 to 30 minutes. An instant-read thermometer should read 200 to 205°F at the center.
Remove from the oven and immediately invert the cornbread onto a rack to cool for at least 10 minutes. Consume hot with butter and a glass of cold buttermilk on the side.