I want to add the depth and nuttiness of browned butter into this yellow cake recipe. I am worried that swapping out the oil for butter will chan...

If you brown then clarify the butter it will be more like oil (butter has some water in it.) I think it will be lovely if you replace the oil with melted browned butter. Just make certain not to overcook it so it remains as moist as possible.

I think the flavor would be delicious with the browned butter, the texture may change a bit, oil tends to make for a very moist cake and butter would change. I am wondering (thinking outside the box here) if you added the 1/3 cup oil as directed and add a tablespoon or two of the browned butter just for the flavor. My recipe calls for 1/3 + 1 tbs and honestly don't think it would would make a difference in the texture instead of the +tbs of oil add browned butter, even 2 tbs would be ok I think. This may not be helping you, sorry. I have never made this cake with butter, if you do make with either all butter or a tbs or two please let me know how it turns out.