Melt butter in a Dutch oven over medium-high heat. Add mushrooms, and cook, stirring constantly, 3 to 5 minutes or until tender. Add green onions, and cook 2 minutes.
Whisk together flour, broth, salt, and pepper in a medium bowl; add to mushroom mixture, stirring well to combine. Bring to a boil, and cook, uncovered, 1 minute. Remove from heat.
Combine sour cream and half-and-half in a small bowl, stirring until well blended. Stir into mushroom mixture, and cook, uncovered, over low heat, stirring often, 12 minutes or until thoroughly heated. Yield: 4 servings (serving size: 1 cup)