For the Cake – Preheat oven to 350F, line a 9-by-9-inch pan with aluminum foil, spray with cooking spray or grease and flour the pan; set aside. In a large microwave-safe mixing bowl, melt the butter, about 1 minute. To the melted butter, add the sugars, egg, yogurt, vanilla, and whisk to combine. Add the bananas, zucchini, and stir to incorporate.
Add the dry pudding mix and stir to incorporate (You are not making pudding; simply add the mix as a dry ingredient). Add the flour, baking soda, salt, and stir until just combined, taking care not to over-mix or cake will be tougher as the gluten will over-develop. Stir in the chocolate chips. Pour batter into prepared pan and bake for about 35 to 38 minutes, or until top is set and golden, the center is not jiggly, and edges have pulled away slightly from sides of the pan. A toothpick inserted in the center will likely not come out completely clean because this is a dense cake and both the gooey bananas and melted chocolate chips may prevent a clean toothpick test. Note – Depending on how moist your zucchini is and the moisture content in your bananas, you may need to bake your cake for a longer duration than I did. Bake until your cake is done and cooked through, given your climate and your set of ingredients. Watch your cake, not the clock, when determining how long to bake it; bake until done. This is a very soft and moist cake, and is not like traditional ‘bakery cakes’ or a typical birthday cake. It is a very soft cake, more like a soft banana bread and it will never become a light, airy, bakery-cake kind of cake, no matter how long you bake it.
Allow cake to cool in pan for at least 30 minutes before slicing and serving. While cake cools, make the glaze.
For the Vanilla Browned Butter Glaze – Brown the butter in a small saucepan on the stovetop, heating over medium heat. Swirl the pan or stir frequently for about 4 to 7 minutes, or until the sputtering, crackling, and foaming has subsided, the butter has browned and has a nutty aroma. Watch it closely so that it doesn’t go from browned and nutty to burnt and inedible.
Or, brown the butter in the microwave in a microwave-safe bowl by heating it on high power, uncovered, for 3 to 5 minutes, until the sputtering, crackling, and foaming has subsided, the butter has browned, and has a nutty aroma as outlined here. The same rules apply in the microwave as on the stovetop; watch it closely and start checking it approximately every 15 seconds starting at about the 3-minute mark, so that it doesn’t go from browned and nutty to burnt and inedible.
Transfer hot butter to medium-sized mixing bowl and allow it to cool for about 5 minutes. Add the confectioners’ sugar, vanilla, and whisk to combine. Based on desired glaze consistency and taste preference, add the cream one tablespoon at a time, until desired consistency is reached, playing with sugar and cream ratios as necessary. (I only use butter, sugar, and vanilla in this glaze, no cream). Drizzle glaze over the top of cake before slicing and serving; or slice, serve, and glaze each piece individually. I use the glaze like butter and spread it liberally on the interior surface of a slice.
Unglazed cake can be stored in an airtight container at room temperature for up to 5 days, or store glazed cake in the refrigerator for up to 5 days. Unglazed cake can be frozen for up to 3 months.