Preheat the oven to 400 degrees F.
Heat a 12-inch cast-iron skillet on the stovetop over medium heat. Add 1/4 cup butter and allow to melt while you begin to prepare the batter. Reserve the melted butter for later use, carefully wipe out the pan and place in the oven to heat.
In a large bowl, combine the cornmeal, flour, baking powder, salt and baking soda. In another bowl, combine the milk, buttermilk, egg and melted butter. Make a well in the dry ingredients and gradually incorporate the wet mixture, stirring until the lumps are dissolved. Fold in the cheese, corn and poblanos.
To bake the cornbread, carefully remove the skillet from the oven and add the remaining 2 tablespoons butter so it sizzles. Coat the pan with the butter and quickly pour the cornbread batter into the skillet. Place the skillet in the oven and bake until golden brown, about 20 minutes.
Remove from the oven and let cool slightly before slicing. Serve warm.
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