Preheat your oven to 350°F (175°C). Grease and flour a 9x13 inch baking pan or line it with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, and nutmeg. Set aside.
In a large mixing bowl, combine the granulated sugar, brown sugar, and vegetable oil. Mix until well combined.
Add the eggs one at a time, mixing well after each addition. Stir in the grated zucchini and vanilla extract.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix. Fold in the chopped walnuts.
Pour the batter into the prepared baking pan and spread it evenly.
Bake for 30-35 minutes, or until a toothpick inserted in the center comes out clean. Allow the cake to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.
If desired, mix the powdered sugar and milk in a small bowl until smooth to create a glaze. Drizzle over the cooled coffee cake before serving.