Preheat oven to 350° and spray a 9×13-inch baking dish with cooking spray.
Season chicken with salt and pepper and dredge in flour.
Heat Vegetable oil in large nonstick skillet. Saute chicken until lightly browned on both sides. (Cook chicken in two batches if necessary.) Transfer chicken to baking dish.
Add 2 tablespoons butter to drippings left in skillet and sauté shallot and mushrooms over medium-high heat until mushrooms are brown. Add tarragon, sprinkle with 1 tablespoon of flour and cook 1 minute. Gradually whisk in half and half. Then add sherry and mushroom soup. Check sauce for seasoning and add salt and pepper to taste. Bring sauce to a simmer and remove from heat.
Pour sauce over chicken, cover with foil and bake 20 minutes.
Top with cheese and broil 1-2 minutes. Sprinkle with parsley and green onion.