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.
Scrape batter into a buttered 8-inch square pan and spread smooth.
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).
Cut bread into squares. Lift from pan with a slender spatula.