Rare Welsh Rarebit

Has anyone made Welsh Rarebit thickened with egg like this? Every other recipe I looked up after trying this one is thickened with a roux. I tried to search for any recipes involving sauce thickened with egg and couldn’t find anything. Needless to say, it did not work. I should have gotten a picture but it was all separated and the cheese quickly thickened back up so it was a mess of thin liquid and globs of cheese. Is there a way to make this work? There has to be…this is from Joy of Cooking, an American book but I figured I’d ask here since it’s a Welsh dish.