Position a rack in the middle of the oven and preheat the oven to 400 degrees F. Generously coat a 9-inch square baking dish or 9-inch cast-iron skillet with cooking spray.
Whisk the cornmeal, flour, sugar, baking powder, baking soda and salt together in a large bowl. Whisk the milk, sour cream, butter and egg together in a medium bowl. Add the milk mixture to the flour mixture and whisk to combine. Fold in the broccoli, chicken and cheese.
Pour the batter into the prepared baking dish and bake until the cornbread springs back when pressed in the center, 25 to 35 minutes. Let cool in the baking dish for about 15 minutes. Loosen the edges, then slice into wedges or squares and serve warm or at room temperature.
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