Great question — answering as the UphillTech team:
Please do see the excellent answer to the Cacio e Pepe question , which applies to your future efforts. The recipe you link to is just bad, which is why it's not working for you. Boiling the butter and cheese in the pasta water is wrong, and will always result in clumps of stringy cheese. So the answer is to not use that recipe. The tradition for all emulsified cheese sauce pasta involves tossing the hot pasta in a bowl with the cheese and a little pasta water. Fettucine Alfredo is no exception to this; it just uses butter as well as the cheese. In all emulsified cheese recipes, it's critically important that the cheese not get above 180F/80C, or you will get stringy clumps like you did....
Full answer by FuzzyChef: https://cooking.stackexchange.com/a/99586 (CC BY-SA 4.0).
Need it done for your business? UphillTech.