Saute onions, and garlic in 1 tablespoon butter until translucent and set aside to cool. Mix beef with cooled onion mixture, eggs, bread crumbs, parsley, and mushrooms. Form into 4-ounce patties and saute in the remaining 2 tablespoons of butter until golden brown on both sides and set aside. In saucepan melt 2 ounces of butter and add flour and cook over low heat for 1 minute until wet sand consistency. Begin to add stock 1/3 at a time whisking and bringing to simmer each time making sure there are no lumps. Add mushrooms and simmer for 1/2 hour. Place patties in a roasting pan and pour gravy over patties and bake in a preheated 375 degree oven for 45 minutes. Garnish with grilled onions and serve.