In a bowl large enough to hold the chicken, mix all the marinade ingredients. Whisk it together until it forms an emulsion, and you can't see any streaks of oil.
Add the chicken to the marinade, and toss to coat thoroughly. Let set for at least 30 minutes, but no more than two hours.
Peel and cut the potatoes into bite-size pieces.
Cut the onion into chunks.
Place the potato, onion, and carrots into a 9" x 13" baking dish.
Remove the chicken from the marinade. Pour the remaining marinade over the vegetables, and toss to coat.
Place the marinated thighs on top of the vegetables.
Bake in a 350℉ oven for about an hour, or until the internal temperature is 165℉.
Mix the butter and flour until it forms a smooth paste.
Remove the chicken from the pan, and stir in the butter / flour mixture (beurre manié). Return the chicken to the pan.