In a large mixing bowl, beat butter and sugar until creamy and smooth (about 2–3 minutes on medium speed).

Introduce the eggs and vanilla extract to the mixture, beating until thoroughly combined and smooth.

In a separate bowl, combine flour, baking soda, and salt. Gradually add the dry mixture to the wet ingredients, mixing until just combined.

Gently fold in the mashed bananas until evenly distributed. Pour the batter into a well-greased 13x9-inch baking pan.

Bake in a preheated 325°F oven for 35-40 minutes, rotating halfway through. Use a toothpick to check for doneness. Let the cake cool completely on a wire rack.

In a small saucepan, melt the butter over low-medium heat until it turns deep golden brown and smells nutty (about 10 minutes). Remove from heat and cool slightly.

Pour the brown butter over powdered sugar in a bowl. Add milk and vanilla extract, then whisk until smooth. If separated, briskly mix until emulsified.

Spread the icing evenly over the cooled cake. Cut into slices and serve.