Preheat the oven to 350°F and grease a 9×13-inch pan; line with parchment if desired. Shred zucchini on the large holes of a box grater and place in a clean towel; press gently to remove only obvious excess water until you have 2 cups packed shredded zucchini.

In a large bowl whisk together flour, baking powder, baking soda, cinnamon, salt, and nutmeg. In another bowl whisk granulated sugar and brown sugar with eggs, then add oil, yogurt, and vanilla and whisk until smooth. Pour wet into dry and stir until just combined, then fold in shredded zucchini and optional nuts.

Spread batter into the prepared pan and smooth the top. Bake at 350°F for 32–38 minutes for a 9×13 pan (or 24–30 minutes for two 9-inch rounds) until a toothpick shows moist crumbs and the top springs back lightly. Cool in the pan 20 minutes, then invert onto a rack to cool completely.

Beat softened cream cheese and butter together until smooth. Add powdered sugar gradually, then cinnamon and vanilla; beat until combined and smooth. If frosting is too soft, chill 10 minutes or add up to 1 tablespoon extra powdered sugar to firm to spreadable consistency.

If using a 9×13 pan, spread frosting evenly over completely cooled cake. If using rounds, level the cakes if needed, spread frosting between layers and over top, and chill briefly to set before slicing.