At least one of the mushroom varieties for this pasta dish should be a richly flavorful one such as shiitake, hedgehog or maitake. Or porcini, of course, if you are fortunate enough to have those. Be sure to follow the cooking instructions, giving the mushroom and green garlic mix a fierce saute to deepen flavors. You have a choice with the pasta quantity. For a mushroom-dense dish, go with 8 ounces; for a more traditional mushroom pasta, 12 ounces is preferable.
Melt butter with olive oil over medium heat in a large skillet. Add mushrooms and saute until they soften, release liquid and then re-absorb it, about 10 minutes.
Stir in green garlic and raise heat to medium-high. Continue cooking until mushroom blend gives off a deeply savory, almost caramelized aroma. Stir in radicchio, cooking briefly until it wilts slightly.* Season to taste with salt and pepper.
Strain pasta, reserving at least 1 cup of cooking water. Add pasta to skillet (or, if there is not room, combine pasta and sauce in the empty saucepan). Simmer a few minutes, stirring gently while adding half the cheese and cooking liquid as needed for a saucy consistency.
Sprinkle servings with remaining cheese.
*The radicchio will darken as it cooks. If you prize that ruby-red color, reserve some to stir in near the end.
Note: An ounce of pancetta, finely diced and sauteed, would be a good addition to this sauce.