Preheat oven to 350ºF. Prepare a 9-inch baking pan with nonstick spray.
Grate zucchini and squeeze out excess liquid - either press into a colander with a spoon, or squeeze zucchini in a clean kitchen towel or paper towels.
In a medium bowl, mix flour, baking powder, baking soda, cinnamon, salt, granulated sugar and brown sugar. Set aside. Next, in a large bowl whisk oil, yogurt, eggs and vanilla. Add dry ingredients to wet, and stir until just combined. Set aside. Gently fold zucchini and walnuts into the batter and pour prepared baking pan Bake at 350°F oven for 25 to 30 minutes or until a toothpick inserted into center comes out clean. Let cool completely in pan on a a wire rack. Frost with Brown Butter Frosting, cut into squares and serve. Leftover cake can be covered with plastic wrap and stored at room temperature for up to four days.
Brown Butter Frosting: In a small saucepan, brown butter over medium heat until fragrantly nutty and deeply browned. Immediately pour it into a medium-sized bowl and let cool 15 minutes. Cream with an electric mixer until fluffy. Add powdered sugar, vanilla, a pinch of salt and cream and beat until smooth. Add additional cream as needed. Spread the frosting evenly over the top of the cooled cake and garnish with a dusting of ground cinnamon.