Preheat oven to 350°F and grease a bundt cake pan extremely well.
Brown the butter in a small-medium saucepan or pot over low-medium heat until it bubbles to brown bits and gives off a nutty aroma. Remove it from the heat, transfer it to a large mixing bowl, and let it cool. (You can set it in the refrigerator to help speed up the cooling)
While the butter is cooling, mash your bananas and, in a separate bowl, whisk together your dry ingredients (flour, baking soda, cinnamon, and salt).
When the butter is about room temperature, whisk the sugars in really well (~2 minutes). Mix in the eggs one at a time, and then the vanilla and sour cream. Stir in the mashed bananas.
Gently stir the dry ingredients into the wet ingredients just until the flour streaks have all disappeared. Then, fold in the chocolate chips.
Evenly pour the batter into the bundt pan and smooth the top with a spatula.
Bake the cake until the top is light brown and a toothpick inserted in the middle comes out clean (about 45-55 minutes).
Let the cake cool in the pan for about 20 minutes. If it looks like it stuck to the sides a bit, carefully run a butter knife around the edges. Then, with a cooling rack on top of the pan (upside-down), flip the cake so that it comes out of the pan and is sitting on the cooling rack.