1. In a bowl, mix flour, cornmeal, 2 tablespoons sugar, baking powder, and salt. In another bowl, beat eggs to blend with buttermilk and 1/4 cup butter. Stir green chilies and cheese into egg mixture. Pour liquids into flour mixture and stir just until evenly moistened.
2. Scrape batter into a buttered 8-inch square pan and spread smooth. In a small bowl, combine chili powder and remaining 1 tablespoon sugar; sprinkle over batter in pan.
3. Bake in a 400° oven until bread springs back when lightly pressed in the center and begins to pull from pan sides, about 25 minutes (20 minutes in a convection oven).
4. Cut bread into squares. Lift from pan with a slender spatula. Serve hot or cool.