Heat the oven to 425°F. Rinse the chicken and pat it dry with paper towels. Cut away any excess fat and tuck the wings behind each breast.
In a small bowl, mash the soft butter, lemon zest, 1 Tbs. of the lemon juice, the mustard, and the tarragon until well blended and smooth. Stir in 1 tsp. of the salt and pepper to taste.
Loosen the skin from each chicken piece to create a pocket. Be careful not to tear the skin or peel it away completely. Spread an equal amount of the flavored butter under the skin of each piece. Use your fingers to massage the butter around so that it spreads out as evenly as possible under the skin. Pull the skin back in place over the butter.
Arrange the chicken in a shallow baking pan that can safely go on the stovetop. (An enamel-coated cast-iron pan such as a Le Creuset or a stainless-steel pan would work well.) Sprinkle with the remaining 1/2 tsp. salt and more pepper. Bake, basting at first with the remaining lemon juice and then the pan juices, until the chicken is browned and cooked through, 55 to 60 minutes.
Remove the chicken from the pan and loosely cover it with foil. Pour off the pan juices into a small measuring cup and let them settle until they separate. Spoon off the fat and discard it. Meanwhile, set the baking pan over medium-high heat and add the chicken stock and 1/2 cup water. Bring to a boil, scraping the bottom and sides of the pan to loosen the browned bits. Continue boiling until the sauce is reduced by half. Add the reserved pan juices, taste, and adjust the seasoning. Spoon a little sauce over each piece of chicken before serving.