Preheat the oven to 375°F. Line a 9×13-inch baking pan with parchment paper.
In a large mixing bowl, beat the butter and sugar until light and fluffy. Mix in the sour cream and eggs, then stir in the mashed bananas and vanilla.
Add the flour and baking soda, mixing just until combined. Fold in the walnuts.
Spread the batter evenly into the prepared pan and bake for 25–30 minutes, or until a toothpick inserted in the center comes out clean. Let the bars cool completely.
To make the frosting, melt 1 stick of butter in a saucepan over medium heat, cooking until it turns golden brown and develops a nutty aroma. Remove from the heat and whisk in the powdered sugar, one cup at a time. Add the vanilla and milk, whisking until smooth.
Spread the frosting over the cooled bars. Let it set before slicing and serving.
Notes
Brown the butter until it's golden and fragrant for the best frosting.
Use very ripe bananas for the sweetest flavor and moistest bars.
Mix just until the flour is incorporated to keep the bars tender.
Substitute pecans for walnuts, or leave the nuts out altogether.
Fold in chocolate chips for an extra indulgent treat.