1Heat the oven to 350°F and arrange a rack in the middle. Coat a 9-by-5-inch loaf pan with butter.
2Melt the measured butter in a small saucepan over medium-high heat. Keep your eye on the pan and cook until the butter solids at the bottom of the pan turn golden brown, about 4 minutes. Transfer to a medium bowl and set aside to cool slightly, about 10 minutes. Add the eggs and buttermilk and whisk to combine.
3Meanwhile, coarsely grate the zucchini, reserving 1/4 cup for garnishing. Add the rest of the zucchini to the butter mixture along with the marjoram or oregano and stir to combine.
4In a large bowl, whisk together the flour, cornmeal, sugar, baking powder, baking soda, and salt to combine and break up any lumps. Add the grated cheese and toss to combine.
5Add the zucchini mixture to the dry ingredients and mix until just combined. Transfer to the prepared loaf pan and smooth the top. Arrange the reserved grated zucchini in a line down the top of the batter. Bake until a tester comes out clean, about 55 to 65 minutes. Let the bread rest in the pan for 10 minutes, then carefully turn out onto a wire rack to cool.