Start by washing and drying your zucchini. Use a box grater to grate the zucchini finely. There’s no need to squeeze out the moisture.
In a large mixing bowl, whisk together the vegetable oil, unsweetened applesauce, and sugar until well combined. Add the eggs one at a time, beating well after each addition. Stir in the vanilla extract.
In a separate bowl, sift together the all-purpose flour, baking soda, baking powder, salt, cinnamon, and nutmeg.
Gradually add the dry ingredients to the wet ingredients, folding gently to combine. Be careful not to overmix. Fold in the grated zucchini and, if using, the chopped walnuts or pecans.
Preheat your oven to 350°F (175°C). Grease a 9x5-inch loaf pan or line it with parchment paper. Pour the batter into the pan and smooth the top with a spatula. Bake for 55-65 minutes, or until a toothpick inserted into the center comes out clean. Let the bread cool in the pan for 10 minutes, then transfer it to a wire rack to cool completely.
While the bread cools, melt the unsalted butter in a small saucepan over medium heat. Continue cooking, swirling the pan occasionally, until the butter foams and then turns a golden brown color with a nutty aroma. Remove from heat immediately.
In a bowl, whisk together the powdered sugar with 1 tablespoon of milk or water. Slowly drizzle in the brown butter, whisking until smooth. Add more milk or water if you prefer a thinner glaze.
Drizzle the brown-butter glaze generously over the cooled zucchini bread. Allow the glaze to set for a few minutes before slicing. Serve warm.