Pat the chicken thighs dry with paper towels.
Heat 2 tbsps. of butter in a non-stick pan. Add the chicken thighs in one layer, skin side down. Cook until the skins are lightly browned. Flip the chicken thighs to brown the underside.
Pour in the peach syrup. Season with salt and pepper. Add the lime zest. Cut the lime into segments. Add to the chicken (the lime will liquefy during cooking). Place the tarragon on top. Bring to the boil, lower the heat, cover and simmer for 30 minutes.
Remove the cover of the pan. Turn up the heat. Add the peaches to the chicken. Cook, uncovered, until almost dry. Turn off the heat. Add the remaining 2 tbsps. of butter, swirling than pan to melt the butter in the residual heat.
Transfer the chicken braised with peaches and lime to a platter. You may want to remove the tarragon stalks. Sprinkle with chopped parsley (optional). Serve hot.