Warm the milk in a saucepan or the microwave until steaming, not boiling. Grate the three cheeses together in a bowl. Bring a large pot of well-salted water to a boil.
Boil the macaroni for 2 minutes less than the package directs, since it will finish cooking in the sauce. Drain and set aside.
Melt the butter in a large, heavy pot or Dutch oven over medium heat. Add the onions and cook for 6 to 8 minutes until soft and translucent. Add the garlic, cayenne, and anchovy fillet. Stir for about 1 minute, mashing the anchovy until it dissolves into the butter.
Pour in the white wine and simmer for 3 to 4 minutes, until reduced by about half.
Sprinkle in the flour and stir constantly for 1 to 2 minutes to cook out the raw taste.
Pour in the warm milk a little at a time, whisking well after each addition to avoid lumps. Simmer gently for 5 to 8 minutes, stirring often, until the sauce coats the back of a spoon.
Take the pot off the heat. Stir in the mustard, then add the cheese a handful at a time, letting each addition melt before the next. Taste and adjust the salt and cayenne.
Fold in the crawfish tails and the drained macaroni. Stir over low heat for 2 to 3 minutes until everything is hot and creamy. Add a splash of milk if it’s too thick.
Serve right away, or for a baked version, transfer to a buttered dish, top with extra cheese and breadcrumbs, and bake at 200°C for 15 to 20 minutes until golden and bubbling.
