Clean mushrooms and remove stems (reserve). Pour a small amount of oil into a container. Using your fingers, rub some oil on the outside of each cap and place one or two drops in the inside center of the cap.
Finely chop stems and toss in bowl with bread crumbs. Add Swiss cheese, grated onion, pepper and toss again. Stir in milk (adding up to 1 TBS more if needed - mixture should hold together). Stuff mixture into mushroom caps, rounding the tops with your fingers. When filled, sprinkle each mushroom with Parmesan cheese. Place mushrooms in baking dish.* Heat oven to 375°. Bake for 15 minutes. 6 servings.
*Mushrooms can be covered and refrigerated at this point. If so, add 3-5 minutes to baking time.