Bring a large pot of water to a boil. Salt the water. Add the pasta and cook according to the package instructions. Drain.
In a large skillet, heat the butter, thyme and garlic over medium heat for 2 minutes. Add the mushrooms and cook until they release their water and begin to brown, about 7 minutes. Season generously with salt and freshly ground black pepper. Add the flour and cook 2 to 3 minutes. Whisk in the milk, cream and mustard. Season with salt and pepper. Bring to a low simmer. Continue whisking until the mixture thickens enough to coat the back of a spoon. Stir in the fontina and Parmesan until melted. Add the pasta and toss to coat.
Serve in individual bowls with a slice of Taleggio on top.