Preheat oven to 450°F. Place cast-iron skillet in the oven for 30 minutes.
Mix the salt and pepper in a small bowl.
Thoroughly rinse the chicken and pat down with paper towels.
Tie the legs of the chicken together and tuck the wingtips behind the breast.
Rub all over with olive oil. Reserving about a teaspoon of the salt and pepper mixture, rub the rest all over the chicken, inside and out.
Using over mitts or a heavy folded towel, carefully place the chicken breast-side up, in the hot skillet in the oven. Roast for 30 minutes.
Turn off the oven and leave the chicken inside it for another 30 minutes, undisturbed. Do not open the door during this time. An instant-read thermometer insert into the breast should read 150°F, the thigh should register 165°F, and the skin should be crispy and nicely browned.
Remove the pan from the oven and place on the stove. Transfer chicken to a cutting board (have a warmed serving platter at the ready).
Skim all but 1 tablespoon of the fat from the pan and discard. Turn the burner to medium and add the shallot, cooking for a minute or two until softened and just starting to brown.
Stir in the stock and mustard, scraping up any browned bits in the pan. Cook for a few more minutes until reduced to about one cup. Remove the pan from the heat.
Cut the chicken into 8 or 10 sections and transfer to the warm platter. Tent with foil to keep warm.
Return the skillet to the burner and, over very low heat, whisk in the butter bit by bit. Stir in the tarragon and lemon juice. Season with some of the reserved salt and pepper if needed. Spoon the sauce over the chicken and serve.
