Place the chicken in a large bowl. Add the olive oil, sage and lemon slices and turn the chicken to coat, scatting the sage and lemon all over the chicken. Cover and refrigerate for 24 hours. Turn the chicken once or twice during the marinating time.
FOR THE SEASONING:
In a small bowl, combine the kosher salt, black pepper and thyme, mix well.
ROAST THE CHICKEN:
Preheat the oven to 425°.
Scatter the carrots, onion and celery in the bottom of a medium to large roasting pan (with enough room so that there's at least 1-2" of room on all sides for the chicken. Add just enough water to cover the bottom of the pan.
Remove the chicken from the marinade and sprinkle all over and inside with the seasoning blend. Take the sage leaves and lemon slices and stuff the chicken cavity with them. Place the chicken on top of the vegetables.
Tuck the wing tips under the chicken. Use a piece of kitchen twine to bind the legs together. Place the chicken on top of the vegetables and place in the oven, feet first. Roast for approximately 1 1/2 - 1 3/4 hours, basting every 20-25 minutes.
Use an Instant Read thermometer to check for doneness. The safe internal temperature is 165° for poultry. Since the chicken continues to cook after its removed from the oven, I usually take it out when it's around 160°. Transfer the chicken to a carving boarding and lightly tent with aluminum foil. Let the chicken rest for 10 minutes before carving. Juices should run clear, not pink.
MAKE THE PAN GRAVY:
Use a slotted spoon to transfer the vegetables to a serving dish and set aside.
In a small bowl mash together the butter and flour to form a paste. Set aside.
Place the roasting pan with the liquids from the chicken and vegetables over a heating element on the stovetop and heat to a boil. Add the vinegar and whisk. Add the chicken broth and resume the boil. When the stock is boiling, whisk in the paste, stirring constantly, until the sauce is well combined and velvety. Add the chopped sage and cook over medium heat for one minute. Taste for seasoning and adjust with salt and pepper if necessary.
TO SERVE:
Carve the chicken and transfer the pieces to a serving platter. Serve with pan gravy and vegetables on the side.