Place the potatoes, carrots, parsnip and shallot in a roasting pan. Drizzle with the olive oil, season with salt and pepper and swirl to coat.
If there are giblets in the chicken, discard them. Place the lemon slices, garlic, thyme and a drizzle of olive oil inside the cavity. Tie the drumsticks together with kitchen twine and tuck the wings under. Pat the chicken dry with paper towels. Rub the skin all over with 3 tablespoons of the butter and sprinkle with salt and pepper. Place the chicken breast-side up on top of the vegetables and pour 1 cup of the chicken broth into the roasting pan.
Place the chicken in the center rack of the oven and bake until the skin is golden brown and the juices run clear when pierced, 1 hour to 1 hour 20 minutes. At around 40 minutes of cooking, add another cup of chicken stock to the roasting pan to keep the vegetables from burning. A meat thermometer should read 165 degrees F when inserted between the leg and the breast. Remove the vegetables from the pan to a platter and transfer the chicken to a cutting board and tent with foil to rest at about 20 minutes before carving.
Place the roasting pan over medium-high heat and add the wine and cook for about a minute, stirring with an angled wooden spoon to release the brown bits from the bottom of the pan. Once it reaches a simmer, add the remaining 2 tablespoons butter and the flour and whisk to combine, cooking for about 30 seconds. Add the remaining 2 cups chicken stock, 1/2 cup at a time, and bring to a boil. Then simmer until the pan sauce has thickened slightly, 4 to 5 minutes. Season with salt and pepper and the juice from the lemon half. Carefully strain the sauce into a serving bowl.
Carve the chicken, transfer to the platter with the vegetables and garnish with the thyme leaves and the remaining lemon half, cut into wedges. Serve with the pan sauce.