Heat the oil in a large skillet with a tight-fitting lid over medium-high heat.
Add both kinds of onions and season with salt and pepper; cook until veggies are tender-crisp, 5 to 7 minutes. Reduce heat to medium; add garlic and mushrooms and cook for 15 to 30 seconds, being careful to not burn.
Add broth, lemon juice and tarragon and cook for 1 minute. Add chicken then reduce heat, cover and simmer for 20 to 30 minutes, turning once, until cooked through.
Serve chicken topped with lemon tarragon sauce and enjoy!