Preheat the oven to 350°F. Liberally grease and flour the inside of a Bundt pan (see Baker’s Notes).
Place the butter in a light-colored sauce pan and melt on the stove over medium heat*. Continue to simmer until the butter becomes frothy and starts to turn golden in color. Stirring to keep the milk solids from burning and sticking to the bottom of the pan, continue to cook the butter until brown, about 8 minutes. When done, it will be nutty and fragrant.
Meanwhile, mash the bananas in a bowl. In a separate bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
Pour the browned butter into the bowl of a stand mixer fitted with the paddle attachment (or in a large bowl with a hand mixer). Allow the butter to cool for about 10 minutes. Add the sugar and mix together on medium speed for a couple minutes. Turn the mixer to low speed and add the vanilla followed by the eggs, one at a time. Add the mashed banana and mix to combine, about 30 seconds. Stop the mixer and scrape down the sides and bottom of the bowl.
Add half of the flour mixture and mix on low speed to combine. Stream in the buttermilk and mix until absorbed. Add the remaining flour. Stop the mixer just before the last of the dry ingredients fully combine into the batter. Add the chocolate chips and chopped chocolate and fold to combine.
Tip the batter into the prepared bundt pan. Use the back of a spoon to smooth, if needed. Bake the cake for 45 to 50 minutes, or until a long, thin knife inserted into the deepest point in the cake comes out dry or with a few moist crumbs (the batter should not appear to be raw).
Allow the cake to cool for 20 minutes before removing the cake from the pan (see Baker’s Notes). Continue to cool completely before adding the glaze.
*Do not boil on high heat or too much water content from the butter will evaporate out