1. In a bowl, mix flour, cornmeal, sugar, baking powder, and salt. In another bowl, beat eggs to blend with buttermilk and 1/4 cup butter. Add corn, basil, and cheese to 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.
3. Bake in a 400° oven until bread springs back when lightly pressed in the center and begins to pull from pan sides, about 30 minutes (25 minutes in a convection oven).
4. Cut bread into squares. Lift from pan with a slender spatula. Serve hot or cool.