Shred zucchini on the large holes of a box grater. (You should have about 4 cups.) Toss with 1/2 teaspoon salt in a colander set over a bowl. Let stand 10 minutes. Squeeze out excess moisture with your hands.
Preheat oven to 400 degrees. Brush a standard 12-cup muffin tin with softened unsalted butter. In a large bowl, combine cornmeal, flour, sugar, baking powder, baking soda (if using), and salt.
In a medium bowl, whisk together buttermilk, butter, and eggs (mixture might appear curdled). Add to flour mixture and stir just until combined (do not overmix). Divide batter evenly among muffin cups.
Bake until a toothpick inserted in center comes out clean, 20 to 25 minutes. Let cool on a wire rack 15 minutes before serving.