Are you absolutely certain that the "grittiness" is caused by the cheese not melting, and not because the sauce is curdling? If you cook it too long or too fast, that is what will happen.
If you must use the Kraft stuff (personally, I think it has no flavour compared to real Reggiano), try melting the cheese on low heat in a very small amount of cream first, before you add it to the main sauce pan. If it's still gritty, either it's curdling or you need to use a better cheese.
According to Cooking The Roman Way by David Downie, Fettucine Alfredo is a traditional Roman recipe called "pasta del cornuti" (cuckold's pasta). What either Alfredo Di Lelio III, or Mario Mozzetti, depending on whom you believe, invented in 1914, was the dramatic tableside preparation of Fettuccine Alfredo, which is what made the dish a hit with visiting Americans in the 20's and 30's.
The tableside preparation is really what makes Fettuccine Alfredo what it is: the hot pasta is tossed with the butter and cheese in front of the diner, and then served to them immediately.
Again, according to Downie, the only ingredients of Fettuccine Alfredo are egg fettuccine, lots of butter, lots of Parmegiano-Reggiano, and (if necessary) a little salt.
Recipes which add cream or milk are Americanized recipes designed to allow restaurants to hold orders of Alfredo for a long time under heat lamps (and turn it into a gooey pasty mess). Italians, from my experience visiting Italy, rarely put cream or milk on pasta (a real Italian could speak up here).
Downie has a fun 3-page digression about the ongoing lawsuits between the Roman families who claim to own the name. It's worth a read.
Alan Davidson, predictably, says nothing about Fettuccine Alfredo. The Glorious Pasta of Italy likewise does not cover the dish.
Wikipedia supports Downie's story, except only attributing Di Lelio, and adding the tidbit that Di Lelio apparently called it "Fettuccine al burro" (fettuccine with butter), and the Alfredo name was appended later when it was copied in the USA. Wikipedia also says butter and cheese only, on fettuccine pasta.
So, to answer your question and the questions asked in the comments:
- Fettuccine Alfredo is an Italian dish, if more popular in the USA than in Italy.
- It is a variation of a traditional Italian dish.
- In its traditional form, it has only egg pasta, butter, and cheese.
- The pasta for Alfredo is egg fettuccine.
Best Answer
This is almost certainly due to cooling. Serving on heated plates will buy you some time. If (at home) you use microwave plate warmers you can place them under the plates after you've used them and the residual heat will keep the plates hot. You need to strike a balance as you don't want it to cook too much after serving.
Cream thickens on cooking (unlike butter which solidifies as it cools). So cooking the sauce for a shorter time might help as well. In fact the recipe I've got for this cautions against overcooking the cream (without saying why).
You don't say how much parmesan you add in proportion to the other ingredients, but you may be onto something with reducing it. Unfortunately this will reduce the flavour, and it's not like you can replace it with a much tastier cheese.
The other thing that's commonly done is adding a little pasta water to the sauce. If you're stirring it together, this can simply be a matter of not draining the pasta too much (this also depends on the shape). If you're spooning it over the top, taking a little of the starchy water just before you drain the pasta, and adding it to the sauce, could be worth a try. Again I don't know your exact quantities (like how big your glass is) but anyway it's a matter for experimenting. A tablespoon or two might be a good place to start. You probably need to put it in slowly while stirring.