Cook long grain and wild rice.
Heat oil in large pot over medium high heat; add chicken, onions, garlic, and mushrooms, and cook for 5 to 8 minutes or until vegetables are tender and chicken is no longer pink.
Add prepared rice, broth, wine, tarragon and thyme, and bring to a boil.
Combine a small amount of evaporated milk and cornstarch in a small bowl, stir until smooth. Stir this into soup mixture with the remaining evaporated milk.
Reduce heat to low, and cook, stirring occasionally, for 8 to 10 minutes or until soup is thickened.
Garnish with green onions if desired.