To make cornbread, combine cornmeal, all purpose flour, sugar, baking powder, baking soda and salt in a large bowl.
In a smaller bowl, beat eggs and oil together. Add buttermilk (or soured milk) and whisk together well. Make a well in the center of dry ingredients and pour in milk mixture. Stir a little to combine then add the melted butter. Stir a little more and add cheese. Combine gently with a spatula without over mixing, then set aside for 30 minutes.
To make filling, heat 1 tablespoon oil in a large skillet on medium high. Add ground beef and cook until all the pink is gone, about 10 minutes.
Add garlic, onion, green pepper and tomato. Stir to combine then add chili powder, cumin, paprika, salt and pepper. Lower heat to medium and cook 6-7 minutes or until onions are clear and spices are toasted. Transfer to a plate to cool slightly.
Preheat oven to 400F. Grease and flour a 9″x13″ baking dish. Spread half of the cornbread batter into pan (or same skillet, if using), then top with meat mixture. Dollop remaining cornbread batter evenly on top, then spread to cover filling.
Bake for 20-25 minutes or until cornbread looks golden brown on top.