How to Make It
Preheat the oven to 350°. Prepare a bowl of ice water. In a medium saucepan of salted boiling water, cook the green beans until crisp-tender, about 3 minutes. Transfer the beans to the ice bath to cool, then drain.
In a medium bowl, combine the yogurt, buttermilk, mustard, lemon juice and tarragon and mix well. Season the dressing with salt.
In a large ovenproof skillet, heat the olive oil until shimmering. Season the chicken with salt and pepper, add to the skillet skin side down and cook over moderate heat until golden and crisp, about 8 minutes. Turn the chicken skin side up and roast in the oven for 10 to 12 minutes, until cooked through and an instant-read thermometer inserted into the thickest part registers 160°.
Add the butter, thyme and garlic to the skillet and cook over moderate heat for 2 minutes, basting the chicken constantly. Let rest for 5 minutes.
Add the green beans to the buttermilk- tarragon dressing and toss to coat. Season with salt and pepper. Serve the chicken with the green beans.