Preheat the oven to 300 degrees.
Loosen the breast and thigh skin of the chicken carefully (don't tear) Stuff under the skin with lemon slices, 8 of the sage leaves, and the parsley.
Season the cavity with salt and pepper Stuff the cavity with the remaining lemon half, 3 sage leaves, 1 thyme sprig, and 2 peeled garlic cloves.
Rub 1 Tbsp of the olive oil over the chicken and season it with salt and pepper to taste. Tie the legs together with kitchen string.
Coat the bottom of a roasting pan with the remaining 2 Tbsp of olive oil. Set the chicken in the pan, breast-side down, and roast for 10 minutes.
Meanwhile, in a small saucepan of boiling water, blanch the Brussels Sprouts for 2 minutes. Drain and let cool slightly. Then cut in half.
Add the Brussels Sprouts to the roasting pan along with the beets, turnips, shallots, carrots, squash, remaining sage leaf, thyme sprig, garlic clove and season with salt and pepper. Roast with the chicken and vegetables for 1 hour.
Add 3/4 cup of the chicken broth to the pan and roast the chicken and vegetables for 20 minutes longer.
Increase the oven temperature to 450 degrees. Turn the chicken breast side up and baste with the pan juices. Continue roasting for about 10-20 minutes longer. (until the bird is browned and temperature in the thickest part of the thigh is 165 degrees.)
Transfer the chicken to a platter and cover it loosely with foil.
Continue roasting the vegetables for another 10 minutes or until tender and slightly browned.
Transfer the vegetables to the platter and cover loosely with foil.
Set the roasting pan on a burner and bring the pan juices to a boil. Add the remaining 1/4 cup of chicken broth and the reserved lemon juice and bring to a simmer over moderately high heat, scraping up the browned bits from the bottom of the pan. Remove the pan from the heat and swirl in the butter.
Season the jus with salt and pepper and then transfer to a warmed gravy boat.
Sprinkle the walnuts over the chicken and vegetables and serve with the jus.