Heat butter in a large sauce pan, swirling to coat the bottom. Add onions, garlic and mushrooms; cook 5-6 minutes. Add in 4 cups of vegetable stock and 2 cups of beef stock, salt, pepper, beef and potatoes bringing to a boil.
Combine remaining cup of stock and flour in a glass jar with a lid. Shake vigorously to combine. Add to soup and bring to a boil. Mixture will begin to thicken. Cook over a low boil for 20 minutes, or until potatoes are easily cut with a fork. Top with green onions and serve.