Preheat the oven to 350 degrees Fahrenheit. Spray a 9x13-inch pan heavily with cooking spray.
In a small saucepan melt the 3 tablespoons of butter over medium heat, stirring continually until butter begins to brown, it will froth first then turn brown and have a strong nutty aroma.
Remove the browned butter from the heat, and transfer it to a large bowl. Add brown sugar, honey, mashed bananas, and vanilla, and mix well. Stir in the buttermilk, eggs, and oil.
In another bowl whisk the flour, salt, and baking soda. Add the flour mixture to the butter mixture and stir until incorporated.
Pour the batter into the prepared pan and bake for 35-45 minutes or until an inserted toothpick comes out clean. Remove from the oven and let it cool completely.
Prepare the frosting. Using a hand mixer, whisk the sugar, brown sugar, and butter until smooth. Add the milk and vanilla. (Add just enough milk so that the frosting comes together and is spreadable)
Spread the frosting on the cooled cake. Let the frosting set before slicing and serving. Store in an airtight container at room temperature.