Heat an ovenproof skillet large enough to hold the chicken over high heat. Add the olive oil and swirl it in the skillet. As soon as you begin to see slight wisps of smoke, carefully place the chicken skin side down in the skillet. Sear the chicken, undisturbed, while reducing the heat little by little to medium, until its skin has turned golden brown and crisp, 5 to 7 minutes. Carefully turn the chicken skin side up.
Put the skillet into the oven and cook until the chicken is deep golden brown and the juices run clear when the thickest part of the thigh is pierced with a skewer, about 10 to 15 minutes, depending on its size. When the chicken is done, transfer it to the plate and keep warm.
Pour off all but a thin layer of fat from the skillet. Add the port, put the skillet over high heat, and reduce until half remains. Add the chicken stock and reduce again. (If using BBQ sauce, you don't have to reduce.) Add the cream, bring it to a boil, and cook for 3 minutes. Stir in the Dijon and Meaux mustards and adjust the seasoning with salt and pepper. Return the chicken to the skillet and sprinkle with parsley and tarragon.