Preheat the oven to 350°F. Spray a 10-inch cast-iron skillet with nonstick cooking spray.
Whisk together the flours, cornmeal, baking powder, baking soda, and salt in a large bowl to combine. Whisk together the brown sugar and melted butter in a small bowl until it forms a thick slurry. Whisk the eggs in another medium bowl. One at a time, whisk the milk, then pumpkin purée, then vegetable oil, and finally the butter-sugar slurry into the eggs. Pour the wet mixture into the dry ingredients and fold gently just until incorporated.
Pour the batter into the prepared pan. Bake for 45 to 50 minutes, until a cake tester comes out clean. Remove from the oven and let cool slightly.
Beat the butter and cinnamon in the bowl of an electric mixer until smooth and creamy. Smear over the warm cornbread.