For the cake:
Preheat oven to 350°F. Butter and flour an 8-inch round pan and line with parchment paper.
In the bowl of a stand mixer fitted with a paddle attachment, combine the flour, sugar, baking powder, baking soda, cinnamon, and salt. Add the butter and mashed banana and mix on low until moistened, then increase the speed to medium high and beat until light and fluffy, about 3 minutes. Scrape the bowl down as necessary.
In a separate bowl, whisk together the eggs, buttermilk, and vanilla extract. Add to the mixer in 2 additions, mixing after each and scraping the bowl down. Fold in the chocolate chips.
Pour into the prepared cake pan and bake until a toothpick inserted into the center comes out clean, 30-35 minutes. Let cool for 10 minutes in the pan, then turn out onto a wire rack to cool completely.
For the icing:
In a small saucepan over medium low heat, melt the butter. Continue to cook, while watching closely, until browned and dark specs appear throughout.
Pour into a bowl and immediately add the powdered sugar and vanilla extract. Add milk to achieve desired consistency. Frost cooled cake immediately.
The cake can be stored, wrapped, at room temperature for several days.
Calories: 393kcal, Carbohydrates: 54g, Protein: 4g, Fat: 19g, Saturated Fat: 11g, Cholesterol: 71mg, Sodium: 139mg, Potassium: 148mg, Fiber: 1g, Sugar: 42g, Vitamin A: 486IU, Vitamin C: 1mg, Calcium: 44mg, Iron: 1mg