In medium bowl, combine beef, lime juice, cumin and salt and toss to evenly coat. Heat 1 tablespoon oil in 10 1/2-inch cast iron skillet. Add meat and cook on high until brown; transfer to plate with juices. Heat remaining tablespoon oil in skillet; add onion and bell pepper. Cook on medium-high until onions begin to brown. Transfer to plate with beef.

In large bowl, combine cornbread mix, eggs and milk; stir until blended. Stir in green chilies. Pour 2 tablespoons oil into the cast iron skillet. Heat until oil is hot; pour 1 1/2 cups of cornbread batter into pan, evenly covering bottom of skillet. Bake at 400 F. for 5 to 8 minutes or until cornbread is set, but not brown on top. Spread beef and vegetables on top of cornbread leaving about a 1/2-inch space all the way around the edge. Top beef mixture with cheese. Top with remaining cornbread batter.