Preheat oven to 500 degrees.
Season chicken with salt and pepper and set aside.
Heat a large oven proof skillet over medium heat. Add vegetable oil, once heated add chicken breasts skin side down.
Cook the chicken until golden brown, about 10-15 minutes.
Flip the chicken, add green garlic (if using), and roast in the oven for an additional 10-15 minutes or until the internal temperature reaches 165°F.
While the chicken is roasting, mix together the Greek yogurt, lemon juice, garlic, and tarragon. Season with salt and pepper to taste.
Serve chicken with Greek yogurt sauce spooned over.