Stir together warm water, yeast, and 1 teaspoon of the sugar in a small bowl or 1-cup glass measuring cup. Let stand 5 minutes.

Stir together flour, cornmeal, baking powder, salt, baking soda, and remaining 3 tablespoons sugar in a large bowl; cut butter and shortening into flour mixture with a pastry blender or 2 forks until crumbly. Add yeast mixture and buttermilk to flour mixture, stirring just until dry ingredients are moistened. Cover bowl with plastic wrap, and chill at least 2 hours or up to 5 days.

Preheat oven to 400°F. Turn dough out onto a lightly floured surface, and knead 3 or 4 times. Gently roll dough into a 1/2-inch-thick circle, and fold dough in half; repeat. Gently roll dough to 1/2-inch thickness; cut with a 2-inch round cutter. Reroll remaining scraps, and cut with cutter. Place rounds with sides touching in a 12-inch cast-iron skillet or on a parchment paper-lined baking sheet. (If using a 12-inch skillet, place remaining biscuits in a 10-inch skillet or on a baking sheet.) Brush biscuits with 3 tablespoons of the melted butter.

Bake in preheated oven until golden brown, 15 to 20 minutes. Brush with remaining 3 tablespoons melted butter, and serve.