Preheat oven to 400 degrees F.
Place mushrooms in a large plastic food storage bag. Drizzle in just enough extra-virgin olive oil to coat ¿ about 1/4 cup should do it. Shake the mushrooms to coat and scatter out onto a cookie sheet. Roast 10 minutes, round-side-up. Season with salt and pepper and flip over. Toss drained quartered artichokes with a drizzle of extra-virgin olive oil, garlic, nutmeg, thyme, grated cheese, and parsley. Generously fill the mushroom caps, mounding the filling up. Top the artichokes with shredded Asiago cheese and return to oven. Cook mushrooms 5 minutes more to melt cheese and set filling. Serve warm.