Making hashbrowns with butter

Your recipe is suspect. Melted butter is the same in chemical composition as softened butter. The two react a bit differently in recipes due to the way that they interact with the other ingredients to impart textures: in short, the creamed butter will hold microscopic bubbles that the fully melted butter will not. Adding egg yolks will help the melted butter incorporate, and probably lead to more cakey cookies (compensating for the flatness you get from melted butter), but the effect is hard to isolate.

As for brown butter, what happens is that as moisture evaporates from the butter, the boiling point increases (similar to what happens when making caramel or candy). As the boiling point of the butterfat increases, the temperature of the butter solids (which consist of lactose and milk proteins) begin to caramelize (the sugars) and undergo a maillard reaction (the protein). Both of these reactions will impart unique flavors to the butter, yielding the distinctive brown butter taste.

As the butter heats, the integrity of the fat is damaged. This occurs to all fats, which is why you can only use a batch of fat for deep frying a few times before it needs to be discarded.

This is how my mom prepares it (she's from Brazil, so it might be a tad different):

1. Cut off all the dark brown skin
2. Wash it and put it into a pot of water and bring to a boil
3. Boil for about 10 minutes (it should still be slightly firm, but not hard)
4. Cut it up into french-fry sized pieces
5. Put the slices in a single layer on a cookie sheet and place in the freezer for about an hour
6. Heat oil, fry them for about 10 minutes (until golden brown)
7. Sprinkle with salt

Steps 1-5 can be done ahead of time and you just pull out what you need from the freezer to fry. I have not personally made these, but I ate hers a lot and they were fantastic!