Preheat oven to 375 °F (190 °C). Line an 8×8-inch baking pan with parchment paper, letting edges hang over sides.

In a large bowl, cream together sugar, sour cream, softened butter, egg, mashed bananas, and vanilla extract until smooth.

Stir in flour, baking soda, and salt until just combined. Spread the batter evenly in the prepared pan.

Bake for 20–25 minutes, until the top is golden brown and bars are set. Remove from oven and let cool slightly, still warm. 

While bars bake, make the frosting: melt the butter over medium heat and continue cooking until it reaches a light amber color and develops toasty brown bits. Quickly transfer to a heat-proof bowl.

Whisk powdered sugar and vanilla into the browned butter until smooth. Add milk as needed to reach a consistency thicker than a glaze but thinner than frosting.

When bars are warm but not piping hot (about 10–15 minutes after baking), pour or spread frosting over the top. The warmth helps it melt in slightly for a soft finish.

Allow to cool for a bit before slicing into 16 bars. Store in an airtight container at room temperature for 3–4 days or in the fridge up to a week.