In a separate bowl, combine flour, baking soda, and salt. Slowly add dry ingredients to wet in 3 small batches. Optional: fold in chopped walnuts.
Pour batter into baking dish, spreading evenly, and bake for 20-25 minutes, or until a toothpick inserted in the center comes out clean. Remove from oven to cool slightly.
In large sauce pan, melt butter until just golden brown in color.
Whisk in powdered sugar one cup at a time, followed by vanilla and milk.
Once mixture has come together and thickened, pour warm frosting over top of banana brownies, using a spatula to spread evenly over the top. Cool completely before cutting into squares to serve.
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