Preheat oven to 450º F. Arrange clean mushroom caps with the open cavity up on a parchment paper lined baking sheet.
Add chopped mushroom stems, sliced onion, and butter to a medium skillet set over medium heat. Cook until the onions are tender, about 7 minutes. Add the beef stock and cook until the onions are glazed and caramelized, about 10 more minutes.
Meanwhile, pulse the bread in a blender or food processor until you have made breadcrumbs. Toss with Stone House Seasoning and fill each of the mushroom caps with the seasoned bread crumbs.
Top the breadcrumbs with the onion mixture and then top with grated cheeses. Bake in the oven for 15 minutes. Remove from the oven and allow to rest on the pan to cool slightly before serving, about 5 minutes. Top with fresh parsley, if using, and serve.
Make-Ahead Instructions: Prepare French Onion Soup Stuffed Mushrooms as directed without baking. Cover the baking sheet and store in the refrigerator up to 3 days. At least 45 minutes prior to time to serve, remove the cover of the baking sheet and bake as directed. The stuffed mushrooms may take an additional few minutes to cook if baking straight from the refrigerator.