Remove the moisture from the chicken with paper towels. Heat up a 12 inch cast-iron or aluminium pan over a medium-high flame. Pour 1 Tbl of oil into and melt 1 Tsp of butter in the pan. Place the chicken in the pan, skin side down, and sear the chicken until well-browned. Turn the chicken over and season the browned side with salt and pepper. Brown the second side. Turn it over and season it. Remove the chicken from the pan and degrease the pan.
Return the chicken to the pan and place it into the hot oven. Roast the chicken for 18-20 minutes, turning twice. Remove the chicken from the pan and degrease the pan. Heat the pan over a medium flame and pour in the wine. Scrape the bottom of the pan to remove the tasty, browned bits. Reduce this wine to a thick, red glaze. Throw in the tarragon. Pour in the heavy cream and stir. Add the cayenne pepper and, if necessary, adjust the seasoning with salt and pepper. Reduce the sauce until it coats the back of the spoon. Turn off the flame and stir in the mustard. Return the chicken to the pan and drench the chicken in the sauce. Move the chicken to a serving platter, then dress the chicken with the sauce.