Preheat oven to 400°F.
Generously season chicken with salt and pepper. To a large skillet set over medium-high heat, add oil. When oil is shimmering, add chicken skin side down and sear until golden brown, about 5 minutes per side.
Transfer the seared chicken to a baking dish and place in the oven to finish cooking through; chicken is done when an instant-read thermometer registers 165°F when inserted in the thickest part of the chicken.
Meanwhile, return the skillet used to sear chicken over medium heat and add shallot and garlic; sauté until garlic is fragrant, about 1 minute.
Add the wine and cook until the wine is almost evaporated, about 2 minutes. Add the broth and the tarragon and continue to cook until the liquid has reduced by half, about 5 more minutes. Reduce the heat to low and stir in the crème fraiche; cook just until heated through and thickened (do not boil the sauce), about 2 minutes. Taste and adjust seasoning with salt and pepper.
To serve: Transfer the chicken to warmed plates and spoon the sauce over the top.
