Preheat: Preheat the oven to 350°F and line a 12-cup muffin pan with liners or grease it well.
Brown the butter: In a saucepan over medium-low heat, melt the butter. Let it cook, stirring often, until it foams, turns golden brown, and smells nutty, about 5 minutes. Watch closely so it doesn’t burn. Pour into a large mixing bowl, scraping in all the brown bits, and let it cool for about 10 minutes.
Add the brown sugar: Whisk the brown sugar into the brown butter until fully combined, 1 to 2 minutes.
Add the bananas: Mash the bananas until smooth, then whisk them into the butter-sugar mixture.
Add the remaining wet ingredients: Whisk in the egg, sour cream, milk, and vanilla until smooth.
Combine the dry ingredients: In another bowl, whisk together the flour, baking soda, and salt.
Fold together: Add the dry ingredients to the banana mixture and gently fold just until no flour streaks remain. Don’t overmix, or the muffins will be tough.
Add the chocolate chips: Gently fold in both kinds of chocolate chips, saving a few to sprinkle on top if you’d like.
Fill the pan: Use an ice cream scoop to divide the batter evenly into the muffin cups, filling each about 3/4 full.
Bake: Bake for 25 to 30 minutes, until golden and a toothpick in the center comes out with a few moist crumbs.
Cool: Let the muffins cool in the pan for 5 minutes, then move them to a wire rack. Enjoy warm or at room temperature!