Melt the butter in the pan over medium heat. Add ⅔ of the can of coconut milk and the tapioca flour, whisking quickly until the liquid thickens. Cook for one minute, Turn off the heat.
Cook pasta in salted water until very firm and even slightly undercooked, a degree or so firmer than al dente. Fresh fettuccine will only take seconds to cook. Drain the pasta.
Transfer the drained pasta to the skillet containing the butter and coconut milk, raising the heat to low and toss the pasta well, lifting it up, turning it from the bottom, coating all the strips with the coconut milk sauce and butter.
Add the remaining ⅓ cup coconut milk, vegan Parmesan cheese, a pinch of salt, a little ground pepper, and a dusting — less than ⅛ teaspoon — of nutmeg. Stir again briefly until the fettuccine is well coated with cream. Taste and adjust with salt and pepper.
Serve immediately from the pan, with additional vegan Parmesan cheese on the side.
Placed in an airtight container, this Alfredo sauce made with coconut milk will keep well in an airtight container in the refrigerator for 3 to 5 days and in the freezer for up to 3 months.