Preheat the oven to 375 degrees F. Set the goat cheese out on the counter to allow it to start softening as you cook the other ingredients.
Slice the radicchio into thin ribbons and soak in a bowl of water. This removes some of the bitterness. Slice the onion into thin half-moons. Slice the garlic thinly.
Add the olive oil to the pan, and sauté onions over medium heat until caramelized and beginning to brown, about 25 minutes. If some of the onions are cooking faster than others and getting too dark, you can add a splash of water to the pan from time to time. This will also soften the onions and help them to caramelize.
Once the onions have been cooking for about 15 minutes, Get a large pot of water started boiling for your pasta. Add enough salt so it "tastes like the sea".
While the onions are cooking, use a box grater to grate the fontina and parmesan cheese. I prefer freshly grated cheese, but you can use pre-grated.
When the onions have caramelized, add the garlic to the pan and sauté until fragrant and just beginning to become translucent, 1 - 2 minutes. Drain the radicchio, then add it to the pan and cook for 3 - 4 minutes until wilted. Add the balsamic vinegar and a pinch of salt and stir to throughly combine.
The pasta water should be boiling by now. Add the noodles and cook until very al dente, 6 - 8 minutes.
While the noodles boil, make the goat cheese sauce. Add the heavy cream and goat cheese to the onion and radicchio mixture over medium low heat, stir contstantly until it just comes to a simmer. Once the goat cheese is completely melted and fully incorporated into the sauce, remove from heat and set aside. Do NOT heat the cream over high heat. This will cause the mixture to curdle. Goat cheese can be fickle.
Reserve one cup of the pasta water, then drain the noodles. Put the drained pasta back into the pot along with the goat cheese mixture, add the pasta water, and stir to combine. Add about ⅔ of the fontina cheese and continue to stir until it's fully incorporated and all melted and gooey.
Transfer the pasta to a buttered oven proof skillet or casserole, top with remaining fontina and the grated parmigiano. Bake for 25-30 minutes until the cheese is browning on top and the sauce is bubbly. Allow to stand for 5 minutes before serving.