Preheat your oven to 350°F (177°C). Grease a 9x5 inch metal loaf pan and line it with a strip of parchment paper long enough to hang over the two long sides. This makes it easy to lift the finished loaf straight out of the pan.
Peel your bananas, break them into chunks, and place them in a large mixing bowl. Mash them thoroughly with a fork until only small lumps remain and the texture looks like chunky applesauce. Set the bowl aside.
Melt the butter in a small light-colored saucepan over medium heat. Once it melts, keep cooking it, stirring occasionally, for about 3-5 minutes. You will hear it crackle and sputter as the water cooks off, then it will go quiet and small golden-brown bits will appear on the bottom of the pan. The butter will smell nutty and toasty. As soon as you see those golden bits, pour the browned butter immediately into the bowl with your mashed bananas and scrape every drop from the pan, since those brown bits carry a lot of the flavor.
Stir to combine, then let the mixture sit for 3-5 minutes to cool slightly before adding your eggs. This step is important. Hot butter poured directly onto eggs can start to cook them before they are incorporated into the batter.
Add the brown sugar to the cooled banana and butter mixture and stir well until combined. Add both eggs and stir until fully mixed in, about 30 seconds. Add the sour cream and vanilla extract and stir again until smooth.
In a separate medium bowl, whisk together the flour, baking soda, baking powder, and fine sea salt. Pour the flour mixture into the banana mixture and stir gently with a spatula or wooden spoon just until no dry streaks remain. Do not overmix. The batter does not need to be perfectly smooth and a few small lumps are fine.
Set aside about 2 tablespoons of the chopped chocolate for the top and fold the rest into the batter with just a few gentle strokes.
Pour and scrape the batter into your prepared pan and smooth the top. Scatter the reserved chocolate chunks over the surface and sprinkle evenly with the flaky sea salt.
Bake on the center rack for 55-65 minutes, until the top is deep golden brown and a toothpick inserted into the center of the loaf in a spot away from any visible chocolate comes out with moist crumbs but no wet batter. If your toothpick hits a melted chocolate pocket, wipe it clean and try a different spot. The very center of the loaf may look barely set, but it will firm up as it cools. Resist checking before the 45-minute mark.
If the top is getting too dark before the loaf is done, loosely tent a piece of aluminum foil over the pan and continue baking.
Let the bread cool in the pan for at least 20 minutes before lifting it out using the parchment paper handles. Transfer to a wire rack and cool for at least another 20 minutes before slicing. Slicing too early can cause the center to look gummy.