Preheat oven to 400ºF. Place mushrooms caps into a bowl. Drizzle with olive oil and toss lightly to coat. Place mushrooms upside-down on a nonstick naking sheet. Bake for 5-8 minutes or until soft and liquid is released. Drain upside-down on paper towel, and set pan with juices aside.
Meanwhile, in a large skillet, saute shallots and garlic in butter for 1 minute. Add chopped portobello stems, oyster and shiitake mushrooms and any liquid left on the sheet pan. Cook and stir for 5 minutes or until mushrooms are very soft and liquid is evaporated. Meanwhile, wipe sheet pan clean and place mushroom caps right-side up on the pan.
Add wine, thyme, salted and pepper to the skillet. Cook and stir, scraping bottom of pan to release bits, until wine has evaporated (about 4 minutes). Remove from heat and add 1/2 cup Mahon, stirring mixture until cheese melts. Divide stuffing between mushrooms caps (about 1 heaping tbsp. for each large portobello, or 1 rounded tsp. per baby bella).
Bake mushrooms for 8-10 minutes or until heated through. Top immediately with remanining cheese and serve. Makes 24.Make ahead: Assemble through step 3 up to one day ahead of time and refrigerate mushrooms on a sheet pan. Bake 10-12 minutes or until heated through, top with remaining cheese and serve.