Melt butter in a large saucepan. Add onion; cover pan and cook over medium-low heat for 8 minutes. Remove cover and add flour. Stir until well blended; cook over medium-low heat for 2 minutes, making sure that the flour does not brown. Combine chicken broth and ½ cup milk; heat until almost boiling. Whisk heated liquid into onion/flour mixture, stirring until smooth and well blended. When mixture begins to thicken, add remaining milk, salt, and pepper; stir until well blended. Cook over medium heat for 10 minutes, stirring often until mixture thickens. Add mushrooms and tarragon. Cook in a covered pan over medium-low heat for 10 minutes. Serve with a tossed green salad or baked potato for a delicious light dinner. Yield: 4 servings.