Brown chicken pieces in the 2 tablespoons hot oil; season with a little salt and pepper. Place chicken pieces in a well greased 9 x 13 inch pan. In a mixing bowl beat eggs; blend in the 1 1/2 cups milk and 1 tablespoon cooking oil. Stir together flour, salt, and tarragon. Add to egg mixture; beat until smooth. Pour over chicken. Bake, uncovered at 350 degrees for 55 to 60 minutes. Meanwhile, in a saucepan, stir together soup, the 1/3 cup milk, mushrooms, and parsley. Heat through, stirring occasionally. Pass the sauce with the casserole when ready to serve.