Shred: Grate the zucchini on the large holes of a box grater (or use the shredding disc of your food processor). You should have about 3 cups of loosely packed shreds. Toss them with 1 teaspoon salt and spread them in a single layer on a plate or in a colander. Let them sit for 10 minutes. You'll see beads of moisture form on the surface — that's the water releasing.
Squeeze: Transfer the zucchini to the center of a clean kitchen towel (or a few layers of cheesecloth). Gather the corners and twist firmly over the sink. Squeeze until you can't get another drop out. You'll be shocked at how much liquid comes out — I always am. Discard the liquid. If you don't have a towel, press the shreds firmly through a fine-mesh strainer with the back of a spoon. It takes a little longer but works fine.
Preheat: Set your oven to 375°F. Place the cast-iron skillet inside to heat while the oven preheats — this helps create that crispy bottom crust. If you're using a baking dish instead, just grease it well with butter or oil.
Mix the dry: In a medium bowl, whisk together the cornmeal, flour, baking powder, baking soda, remaining ½ teaspoon salt, and a few grinds of black pepper.
Mix the wet: In a large bowl, whisk the eggs until they're light and frothy — about 30 seconds. Whisk in the milk, sour cream, and melted butter until smooth.
Combine: Add the dry ingredients to the wet mixture and stir with a wooden spoon just until no streaks of flour remain. A few lumps are fine — overmixing makes cornbread tough.
Fold in the zucchini, cheese, and green onions: Add the squeezed zucchini, ¾ cup of the shredded cheese, and the green onions or chives. Fold gently with a spatula until everything is evenly distributed. The batter will be thick and slightly shaggy — that's normal.
Bake: Carefully remove the hot skillet from the oven (use an oven mitt — it's screaming hot). Add a pat of butter and swirl to coat the bottom. Scrape the batter into the skillet and spread it evenly. Sprinkle the remaining ¼ cup cheese on top. Bake for 35–40 minutes, until the top is golden brown and a toothpick inserted in the center comes out clean or with just a few moist crumbs. The edges should be pulling away from the skillet slightly when it's done. If the top is browning too quickly, cover loosely with foil for the last 10 minutes.
Rest: Let it cool in the pan for at least 10 minutes before slicing. This is the hardest part — the whole kitchen smells incredible — but it's important. The cornbread finishes setting as it cools. Cut too soon and it'll crumble.