In a large mixing bowl, beat butter and sugar until creamy.

Add eggs and vanilla, beating until combined.

In a large bowl, combine flour, baking soda, and salt.

Add flour mixture to mixing bowl, beating until just combined. Do not overmix.

Fold in mashed banana.

Pour batter into a well-greased 13x9-inch baking pan.

Bake at 325 degrees for 35-40 minutes, rotating the pan 180 degrees halfway through baking, or until a toothpick inserted near the center comes out clean. The cake will brown significantly on top - this is okay! It is not burning (provided you have checked the middle with a toothpick). It will be very brown at the edges and on top, but not burned.

Remove from heat, and allow to cool completely on a wire rack.

Meanwhile, prepare brown butter icing.

Place the powdered sugar in a medium bowl, and set aside.

Melt butter in a small saucepan over low-medium heat, stirring frequently, until it foams, the foam dissipates, it turns a deep golden brown, and develops brown flecks and a nutty fragrance, about 10 minutes or so. Toward the later stages, around the time it begins to foam, watch the brown butter very closely. It can go from 'brown' to 'burned' very quickly.

Remove from heat, cool 2-3 minutes, and pour over powdered sugar.

Add milk and vanilla, and vigorously whisk the mixture until smooth. If the icing seems to seize or separate, you just reeeeeeally mix it to emulsify it. You can use an electric mixer, if preferred. Don't give up thinking it's ruined, it just needs to be emulsified, so give it a serious mixing! Icing should be thin enough to easily spread, but not runny or drippy. Add additional milk or powdered sugar, as needed, to achieve desired consistency.

Ice the cake. Cut and serve.