To a medium pot over low heat, add the milk, onion, black pepper, nutmeg, and cloves. Bring to a simmer, then immediately remove from the heat. Set aside to steep for 20 minutes.
Set a fine mesh sieve over a large bowl, then strain the infused milk into it, discarding any solids. Set aside.
Rinse out the pot then set it over medium-low heat and add the butter. When the foam begins to subside, whisk in the flour. Cook the roux, stirring continuously, until it smells slightly toasty and is just begining to stick to the bottom of the pot, about 5 minutes. Slowly begin whisking the infused milk into the roux a little bit at a time. Once all of the milk has been added, turn the heat up to medium-high, bring to a full boil, then turn the heat back down to maintain a simmer and cook, stirring continuously, and until the sauce has thickened, about 5 minutes more. Turn the heat to low and begin slowly adding the fontina, gouda, and Gruyère cheeses, stirring frequently, until completely melted and incorporated. Remove from the heat, season to taste with salt, then cover the surface of the sauce with a sheet of plastic wrap to prevent the formation of a skin. Set aside at room temperature while you boil the pasta.
Preheat the oven (with one of its racks positioned in the center) to 475ºF. Fill a large stockpot two thirds of the way full with water and bring to a boil over high heat. Season the water generously with salt, then stir in the pasta and cook, stirring frequently, until just barely tender about 4 minutes.
Meanwhile, fill a large bowl halfway with ice water and place it by the sink. Set a large colander in the sink, strain the pasta, then transfer immediately to the ice water bath. Once cooled, strain the pasta again and return it to the stockpot. Stir in the reserved cheese sauce, then transfer to a 9-by-13-inch baking dish. Transfer the dish to a large rimmed baking sheet to catch any spills and sprinkle the Parmigiano evenly over the top. Transfer to the oven to bake until the surface is golden brown, the pasta is cooked through, and the sauce is beginning to bubble, about 15 minutes. Serve hot.