Bring a large pot of lightly salted water to a boil.
Add the butter to a large skillet over medium heat and add the mushrooms and garlic. Cook until browned, about 5 minutes, then add the wine and cook until evaporated.
Meanwhile, add the pasta to the boiling water and cook according to the package directions, leaning toward the al dente side. Do not overcook; this will take 1 to 2 minutes!
Add the cream to the skillet with the mushrooms. Drain the pasta (reserving some of the pasta water) and add the pasta to the mushrooms, along with the fontina, Parmesan, Romano, goat cheese, 1/2 teaspoon salt and some pepper. Stir gently to combine, adding hot pasta water as needed for consistency. Don't overmix; if there are little clumps of cheese here and there, it's fine! Taste and add more salt if needed. The mixing stage should happen very quickly.
Turn the pasta out into a large serving bowl. Sprinkle with parsley and serve immediately.
You can use the 4 cheeses above, or whatever 4 cheeses you'd like!