Preheat oven to 350°F. Prepare Bundt pan by buttering & dusting with flour, or spraying with non-stick spray.
In a small saucepan, melt butter over medium heat. Bring it to a bubble and allow it to simmer until it starts to turn light brown and smells nutty, about 5-7 minutes.
Pour brown butter into a cup with a spout or a bowl; set aside to cool.
Meanwhile, sift together the salt, flour and baking soda in a small bowl.
In a stand mixer (or a large bowl), whisk together sugar and cooled butter until smooth. Add in the eggs, mashed banana and vanilla extract and beat until everything is well-incorporated.
Add the flour mixture into the wet ingredients. Mix until just incorporated. Stir in chocolate chips until evenly distributed.
Pour the batter into the prepared baking pan and bake for about 50 minutes, or until a knife or skewer inserted in the middle comes out mostly clean.
Remove the cake from the oven and set it aside to cool for 15-20 minutes. Once cooled, invert over a rack or pan to remove cake.
Dust with powdered sugar if desired.