Heat the oil in a large skillet with a tight-fitting lid over medium-high heat. Add onion, season with salt and pepper then cook until soft.
Reduce skillet heat to medium and add garlic; cook for 15 to 30 seconds, being careful to not burn it.
Add broth, juice, vinegar and tarragon; cook for 1 minute.
Add chicken and mushrooms then reduce heat, cover and simmer for 20 or 30 minutes, turning chicken once, until cooked through.
Serve chicken topped with mushroom tarragon sauce and onions; enjoy!