Trim all visible fat from chicken breasts. If chicken breasts are quite thick, pound them with a meat mallet until they are about 3/4 inch thick. Season with salt and pepper.
Heat 1-tablespoon canola oil and butter in a large frying pan over medium-high heat. Add the chicken (do not crowd). Saute until chicken is cooked through and well-browned on both sides, about 10-13 minutes. Set aside.
Heat 1-tablespoon canola oil in a pan over medium-high heat. Add the mushroom and saute until browned on both sides, about 7-8 minutes. Add the green onion and saute for 1 minute. Add the garlic and saute for 1 minute. Then add the chicken broth and cream. Bring to a boil and let simmer over medium heat until slightly thickened, about 7-8 minutes. Add the mustard and tarragon. Season with salt and pepper, stir well. Return the chicken to the pan and cook for 2-3 minutes.
Transfer to a serving dish and garnish with fresh tarragon. Serve with Mashed Potato or salad.