Preheat your oven to 400°F and position a rack in the middle.
In a 10-inch cast iron skillet over medium heat, add the neutral oil. Add the diced onion, celery, and bell pepper with a generous pinch of salt. Cook, stirring occasionally, until soft and translucent, about 8 minutes. Do not let them brown.
Add the tomato paste and stir constantly for 2 minutes until it darkens and begins to stick to the pan.
Add the garlic, cumin, smoked paprika, chili powder, and black pepper. Stir for 30 seconds until fragrant. Pour in 1/2 cup of the chicken stock and scrape up the browned bits from the bottom of the pan.
Add the shredded chicken, black beans, and the remaining 1/2 cup of stock. Simmer for 5-7 minutes until the liquid thickens slightly. Taste and adjust salt and pepper.
While the filling simmers, make the cornbread topping: In a large bowl, whisk together cornmeal, flour, sugar, baking powder, baking soda, and salt. In a separate bowl, whisk together buttermilk, egg, and melted butter. Pour the wet into the dry and stir until just combined. Fold in the cheddar cheese and green chiles.
Drop spoonfuls of the cornbread batter over the hot filling and spread gently to the edges. Some filling should peek through.
Transfer the skillet to the oven and bake for 22-25 minutes, until the cornbread is golden brown and a toothpick inserted into the center comes out clean. Let rest for 5 minutes before serving.