For the roasted chicken, preheat the oven to 425°F.
Melt 1 tablespoon of the butter in a skillet. Add the diced carrots, onion and celery and cook over moderate heat until softened. Stir in the herbs and set aside.
Wash the chicken rapidly inside and out with hot water and pat thoroughly dry. For easier carving, cut out and discard the wishbone. Pull the neck skin up over the breast and secure it to the back with a toothpick. Salt and pepper the cavity and spoon in the cooked vegetables, a handful of parsley stems and celery leaves and the lemon slices.
Massage the chicken all over with 1 tablespoon of the butter, then truss it. Alternatively, tie the ends of the drumsticks together and tuck the wings under the body.
Choose a flameproof roasting pan that is about 1 inch larger than the chicken. Salt the chicken all over and set it breast up on a rack in the pan. (Thoroughly wash all surfaces and utensils that have been in contact with the raw chicken.)
Roast the chicken in the oven for about 1 hour and 15 minutes, as follows: At 15 minutes, brush the chicken with the remaining ½ tablespoon of butter. Scatter the sliced onion and carrots all around. Reduce the oven temperature to 350°F. At 45 minutes, brush the lemon juice over the chicken. If necessary, add ½ cup of water to the vegetables to prevent burning. At 60 minutes, baste with the pan juices. Test for doneness: The drumsticks should move easily in their sockets; their flesh should feel somewhat soft. If not, continue roasting, basting and testing every 7 to 8 minutes, until an instant-read thermometer registers 165°F.
Spear the chicken through the shoulders and lift to drain; if the last of the juices run clear yellow, the chicken is done. Let rest on a carving board for 15 minutes; discard the string.
Spoon all but 1 tablespoon of fat from the juices in the pan. Add the stock and boil until lightly syrupy, 5 minutes. Strain; you will have just enough to bathe each serving with a fragrant spoonful.
For the glazed carrots, while the chicken cooks, put the first five ingredients in a pan on the stove and heat to melt the butter. Add the carrots to the pan and cover. Boil slowly until tender, about 20 to 30 minutes. Serve with the chicken.
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