Combine 1 tablespoon chicken broth and cheese in a small bowl, stirring with a whisk until blended. Sprinkle chicken with salt and pepper.
Melt butter in a large nonstick skillet over medium-low heat. Add onion, celery, carrot, bell pepper, and garlic; sauté 3 minutes. Add chicken; cook for 2 minutes. Stir in 3 tablespoons broth; cover, reduce heat, and simmer 15 minutes or until chicken is done.
Stir in cheese mixture and tarragon until blended. Serve with rice.