Has anyone browned the butter?

DO IT!!!! I'm right beside you in the brown butter camp! Just made small rum cakes and browned the butter for the rum syrup that is poured over the finished cakes before they are turned out of the pan. Also I swap in browned butter for regular butter any time I can in equal amounts, usually in baked goods. I do chill the browned butter to solidify if the recipe calls for room temp or chilled butter. I know in browning butter, you lose a small amount of water that boils off in the browning process, but I haven't had a recipe yet that didn't turn out due to the lack of that small amount of water.

While I haven't made this specific recipe I have baked similar cakes (including financiers) and brown butter would likely work very well.

Best of luck.