Clean mushrooms and remove stems. Mince the stems and reserve.
Add oil and butter to a non-stick saute pan and saute the yellow onion (or shallots, whichever you use) for 3 - 4 minutes.
Add green onion and mushroom stems, season with salt and pepper to taste, and saute for 2 - 3 minutes more, then turn off heat.
Add remaining ingredients (except for shredded Swiss cheese) to the onion-mushroom mixture and stir well to thoroughly combine. You want the mixture to be moistened, but not dripping extra liquid, so adjust bread crumbs or cream amounts as needed to get the right consistency (like a thick paste).
Place the mushroom caps in a baking dish and fill with stuffing (heap it on pretty good, about 1 heaping Tablespoon per mushroom cap). If you have leftover stuffing, just put in in the baking dish around the mushrooms.
Top stuffed mushrooms with shredded Swiss cheese and bake at 425 degrees Farenheit for 20 minutes.
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