Preheat oven to 350 degrees F. Line two baking sheets with parchment paper or silat; set aside.
For the cookies: Using your stand or hand mixer, mix together the butter, sugars and salt until light and fluffy. Add in the vanilla and bananas.
Mix in the eggs, one at a time, making sure that each egg is incorporated before adding the next.
Whisk together the flour, oats and baking soda, then with the mixer on slowly add until everything is combined. Ladle the sides and bottom of the bowl.
Scoop roughly 2 tablespoons on prepared baking sheet. Bake 12 to 14 minutes, or until tops spring back and edges are lightly golden brown. Let sit on the baking sheet for 5-10 minutes then transfer to a cooling rack.
Let cookies cool completely before frosting.
For the Frosting: Using a small saucepan, add butter over medium heat. Brown butter, making sure to stir constantly for about 3- 5 minutes. Make sure you watch your butter because it can go from golden brown to burnt in seconds. Let cool 5 minutes before making the frosting.
Add the powdered sugar, browned butter and vanilla extract to a bowl. Whisking together, add 1-2 tablespoons of milk, until frosting is smooth and creamy. You want the frosting to be thick. If you need to add more milk, add until desired constancy.
Spread frosting on top of cookies and let sit 10 minutes, so the frosting can harden.
Serve and enjoy!