Preheat the oven to 325°F (163°C).
Remove the turkey giblets and pat the turkey dry with paper towels.
In a small bowl, combine the softened butter, salt, pepper, rosemary, and thyme.
Rub the butter mixture all over the turkey, including under the skin where possible.
Stuff the turkey cavity with the quartered onion, lemon halves, and smashed garlic cloves.
Place the turkey breast-side up on a rack in a large roasting pan.
Pour 2 cups of chicken broth into the bottom of the roasting pan.
Roast the turkey in the preheated oven for about 3 hours and 30 minutes, or until the internal temperature reaches 165°F (74°C) in the thickest part of the thigh, basting occasionally with pan juices.
While the turkey roasts, prepare the mashed potatoes: place the peeled and cut potatoes in a large pot and cover with cold water.
Bring to a boil over high heat, then reduce heat to medium and simmer until potatoes are tender, about 15-20 minutes.
Drain the potatoes and return them to the pot.
Add milk, butter, salt, and pepper to the potatoes and mash until smooth and creamy. Keep warm.
Toss the carrots and parsnips with olive oil, salt, and pepper.
About 45 minutes before the turkey is done, spread the vegetables on a baking sheet and roast in the oven until tender and caramelized, about 40-45 minutes.
When the turkey is done, transfer it to a cutting board and tent with foil. Let rest for 20 minutes.
To make the gravy, pour the pan drippings into a fat separator or skim off excess fat.
Place the roasting pan over medium heat on the stove, add the flour to the pan drippings, and whisk continuously to form a roux, cooking for 2 minutes.
Slowly whisk in 2 cups of chicken broth and continue whisking until the gravy thickens, about 5 minutes.
Season the gravy with salt and pepper to taste.
Carve the turkey and serve with mashed potatoes, roasted vegetables, and gravy.