Preheat oven to 375 degrees.
In a large bowl, whisk together flour, cornmeal, baking powder, baking soda, and salt. With a pastry cutter or your fingers, cut in 1/2 cup cold butter until mixture resembles coarse crumbs with a few pea-size pieces remaining. In a small bowl, whisk together buttermilk and 3 tablespoons honey. Add to flour mixture and stir with a fork until dough just comes together. (Dough will be slightly sticky; do not overmix.) With two large spoons, drop 10 mounds of dough, 2 inches apart, on two parchment-lined baking sheets.
Bake until biscuits are golden, 15 to 20 minutes, rotating sheets halfway through. Transfer biscuits on parchment to wire racks and let cool. Meanwhile, in a small bowl, mix 1/4 cup butter and 2 tablespoons honey until combined. Serve honey butter with biscuits.