Preheat the oven to 350°F (175°C). Grease a 9x5-inch loaf pan or line it with parchment paper.
In a large bowl, combine the grated zucchini, granulated sugar, brown sugar, eggs, vegetable oil, and vanilla extract. Mix until well combined.
In another bowl, whisk together the flour, baking soda, baking powder, ground cinnamon, and salt.
Gradually add the dry ingredients to the wet ingredients, stirring until just combined. Be careful not to overmix. If using, fold in the chopped nuts.
In a small bowl, mix together the brown sugar and ground cinnamon for the swirl.
Pour half of the batter into the prepared loaf pan. Sprinkle half of the cinnamon swirl mixture over the batter. Pour the remaining batter on top, then sprinkle the rest of the cinnamon swirl mixture over the top. Use a knife to gently swirl the mixture into the batter.
Bake for 50-60 minutes, or until a toothpick inserted into the center comes out clean. Allow the bread to cool in the pan for about 10 minutes before transferring it to a wire rack to cool completely.
While the bread cools, prepare the glaze by whisking together the powdered sugar, milk, and vanilla extract until smooth. Drizzle the glaze over the cooled bread.