Preheat oven to 400°F.
Place mushrooms in a large plastic food storage bag. Drizzle in just enough EVOO 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 EVOO, 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.