Preheat oven to 350°F (175°C). Grease a 9x5-inch loaf pan.
Place grated zucchini in a colander with a pinch of salt; let sit for 10 minutes, then squeeze out moisture.
In a large bowl, mix flour, baking powder, baking soda, and salt.
Stir in zucchini, cheddar, Parmesan, and basil.
In another bowl, whisk eggs, buttermilk, and melted butter.
Combine wet and dry mixtures, stir until just mixed.
Pour into loaf pan, smooth top.
Bake for 55–60 mins, or until a toothpick comes out clean.
Let cool 10 mins in pan, then transfer to a rack.