Preheat oven to 350°F. Rinse the chicken and dry it well.
Stuff chicken cavity with herbs, lemon halves and garlic. Drizzle bird with oil and rub it all over; season with salt and pepper. Place chicken on a roasting rack set on a baking sheet. Roast for 1 hour and 45 minutes, until internal temperature reaches 160°F.
While the chicken roasts, place Brussels sprouts, squash, shallot and garlic in a large bowl and toss with the garlic, thyme, lemon juice, oil and honey. Season with salt and pepper. Place vegetables on a baking sheet and roast alongside the chicken for 20-30 minutes, until slightly caramelized and tender.
When the chicken is done, place drippings in a saucepot and add flour. Over low heat, stir to cook the flour, a minute or so, then add the stock and the demi glace. Bring to a slight simmer and reduce slowly for 3 minutes.
Serve chicken on a platter surrounded by the roasted veggies with the sauce on the side.
For the Parsnip Purée, place the parsnips in a pot with the milk. Bring to a boil and cook until tender. Strain parsnips in a colander, reserving the milk.
Purée parsnips in a vita prep or stand mixer until they reach a mashed potato-like consistency, using the reserved milk to thin them out if necessary. Season with salt and pepper to taste.