Prepare rice according to package directions; set aside.
Heat oil in a large Dutch oven over medium-high heat. Add chopped onion and next 4 ingredients (onion through mushrooms), and sauté for 6 minutes or until onion is tender. Lightly spoon the flour into a dry measuring cup, and level with a knife. Stir the flour, tarragon, and thyme into the onion mixture, and cook for 1 minute, stirring frequently. Add 2 cups water, sherry, broth, and evaporated milk; bring mixture to a boil. Reduce heat, and simmer for 20 minutes or until slightly thick. Stir in cooked rice and chicken; cook for 10 minutes or until thoroughly heated.