Preheat oven to 350F. Butter the bottom and sides of an 8×8 inch metal baking pan. Put a long piece of parchment paper in the bottom of the pan, letting the parchment extend up two sides of the pan and overhang slightly on both ends. Place another piece of parchment paper in the same manner, perpendicular to the first sheet. (This will make it easy to remove the bars from the pan after they have baked.) Butter the parchment.
In a large microwave-safe mixing bowl, melt the butter. Whisk in sugars, egg, yogurt, and vanilla until combined. Add the bananas and zucchini; stir until combined.
Add the dry pudding mix to the banana mixture and stir until combined. Add the flour, baking soda, salt, and stir until just combined (do not over-mix). Fold in the chocolate chips. Pour batter into prepared pan and bake for about 35 to 38 minutes, or until golden and edges have pulled away slightly from sides of the pan. Transfer to a wire rack and allow cake to cool in pan for at least 30 minutes before slicing and serving. While cake is cooling, make the glaze.
To make the glaze, in a 10-inch skillet over medium-high heat, melt butter. Continue cooking, swirling pan constantly until butter is dark golden brown and has nutty aroma, 1 to 3 minutes. Remove from heat and transfer to a large heatproof bowl and allow it to cool for 5 minutes. Add the confectioners’ sugar and vanilla; whisk to combine. Based on desired glaze consistency and taste preference, add the milk 1/2 tablespoon at a time, until desired consistency is reached.
