There's Alfredo and there's Alfredo. And now Al Fredo and Al'Fredo?With all due respect, I disagree that the recipe you posted is an Alfredo sauce.
Alfredo's was (and still is) a restaurant in Rome that got famous as a result of Mary Pickford and Douglas Fairbank's patronage. They were two of the most famous people in the world when they got married, and their wedding didn't exactly lack publicity. On their honeymoon, also well publicized, they stayed in Rome and ate at Alfredo's frequently. Their favorite dish famously was something the owner, whose first name was Alfredo, made for them -- which was his own variation on fettuccine al burro and which he called, not unsurprisingly, Fettuccine Alfredo.
According to both his descendants and to the people who now own the restaurant, Alfredo made the sauce by very thoroughly creaming a lot of butter with a lot of cheese, then adding a little of the pasta cooking water, then adding (just cooked) fresh pasta on top and tossing. Alfredo supposedly got the ideas of creaming the cheese and butter so completely and of using fresh pasta (which wasn't very common in Italy until recently) because during his wife's pregnancy it was the only way she could keep it down. Going beyond morning sickness and on to other things: At Alfredo's, the tossing was done with solid gold spoons which Mary and Doug gave him as tokens of their regard. True dat.
As a result of the notoriety, Fettuccine Alfredo became immensely popular in the U.S. of A., and everyone and their brother tried to recreate the dish, or put their own spin on it. Cream and sometimes even an egg were added to the butter and cheese. Sometimes cream replaced the butter. However, at Alfredo's the dish continued to be made in the same way with the only allowable option being the choice of spinach fettuccine.
But stock? And almost no cheese? Whatever it is may be good, and deserves a name no doubt. But, sorry. I don't care what Portofino in Acapulco calls it. Alfredo is taken and your variation wanders too far to share the name.
Here's the original:
3/4 cup butter, room temperature
3/4 cup, well aged, finely grated Parmagiano
1 pound fresh fettuccine
Cream the butter and cheese together until completely smooth. About 3 minutes in a mixer. About 5 - 10 by hand, depending on the hand. Place the butter in the bowl in which the pasta will be mixed and from which it will be served.
Cook the pasta in several gallons of well salted, boiling water to which a little olive oil was added . Fresh fettuccine takes about 3 minutes, four at most. Drain the fettuccine. While it is draining, add 1/4 cup of the pasta water on top of the butter and cheese mixture. Add the pasta immediately. Bring the bowl to the table and toss with a salad fork and spoon. Serve with the tossing utensils. Pass additional grated cheese.