Pre heat oven to 350 degrees.
Grease a 9 x 13 inch baking dish. Line with parchment paper.
In the bowl of a stand mixer with a paddle attachment, beat sugar and butter together until smooth. Add in the yogurt, vanilla and eggs and beat on low until fully incorporated.
In another medium size bowl, mix the flour, nutmeg and baking powder.
Fold the flour mixture into the sugar and butter mixture with a scraper until just blended. Fold in the mashed bananas and walnuts.
Pour the batter into the prepared baking dish and bake at 350 degrees for 30 minutes until a cake tester inserted comes out clean.
When the cake has only about 15 minutes left to bake, make the brown butter icing.
Melt the butter in a large skillet until foamy. The butter will begin to bubble and pop. This is normal. On low heat, slowly swirl the butter in the pan until brown bits begin to form on the bottom of the pan. Remove from heat if butter is browning too quickly.
Do not stir with a spoon- just swirl .
Off heat, gently spoon the powdered sugar, vanilla and milk into the browned butter and whisk until smooth and glossy.
If the icing is not pourable, add more milk to loosen it.
Remove the baked cake from the oven and pour the icing over the cake while still hot.
If you wait to ice the cake, the icing will set up and not pour.
Be certain to ice the cake while hot.
Let cake set up for 30 minutes in the pan.
Remove and slice into squares with a serrated knife. Leftovers may be stored in an airtight container.
