Preheat oven to 350F and adjust oven rack to middle. Line a 9x13 baking pan with parchment paper. Set aside.
In a small saucepan over medium heat, melt butter until a deep golden brown and fragrant. Set aside 2 tablespoons in a medium bowl for glaze and transfer remaining to a large bowl.
In a large bowl with an electric beater or whisk, beat sugar and eggs well 1-2 minutes until eggs are slightly thickened. Beat in mashed banana, browned butter, yogurt, and vanilla until smooth.
In a separate medium bowl, combine flour, baking soda, salt, and cinnamon. Gently fold half of flour mixture into liquid with half of buttermilk and zucchini, then repeat with remaining flour mixture, buttermilk, and zucchini until batter is smooth. Fold in toasted walnuts if using.
Scrape batter into prepared pan and smooth top. Bake cake at 350F 35-40 minutes until a toothpick inserted in middle comes out clean. Cool completely on a wire cooling rack.