Bhatsoy

I make a delicious 'sauce' that mixes melted unsalted butter and shoyu. One can fry vegetables in it, mushrooms are particular delicious, and it can even be poured over rice or ramen for a rich buttery taste. It's not my idea but I have no idea from where I first heard about. I assume it's Japanese. Does anyone know of its origin and most importantly, what it might be called?

If it doesn't have a name would anyone suggest a suitably cool Japanese name for it?

バター醬油 Bata-shōyu. Very common. It's my wife's favorite way to cook steak.