Preheat the oven to 325°F (165°C).
Remove the turkey giblets and pat the turkey dry with paper towels.
Rub the turkey all over with olive oil, then season generously with kosher salt, black pepper, dried thyme, and dried rosemary.
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 15 minutes, or until the internal temperature reaches 165°F (74°C) when measured in the thickest part of the thigh.
Baste the turkey every 45 minutes with pan juices, adding more chicken broth as needed to keep the pan moist.
While the turkey roasts, prepare the mashed potatoes: place the peeled and chunked potatoes in a large pot and cover with cold water. Bring to a boil and cook until tender, about 15-20 minutes.
Drain the potatoes and return them to the pot. Add the whole milk and 4 tablespoons of butter, then mash until smooth. Season with salt and pepper to taste. Keep warm.
Toss the carrots and green beans with olive oil, salt, and black pepper. Spread them on a baking sheet.
About 30 minutes before the turkey is done, place the vegetables in the oven and roast until tender, about 25-30 minutes.
When the turkey is done, transfer it to a cutting board and tent loosely with foil. Let rest for 20 minutes.
To make the gravy, pour the pan drippings into a medium saucepan through a fine mesh strainer, discarding solids. Let the fat rise to the top and skim off 3 tablespoons of fat.
Place the saucepan over medium heat and whisk in the flour to make a roux. Cook for 2 minutes, whisking constantly.
Gradually whisk in the remaining 2 cups of chicken broth and the reserved pan juices. Bring to a simmer and cook until thickened, about 5 minutes. Season with salt and pepper to taste.
Carve the turkey and serve with mashed potatoes, roasted vegetables, and gravy.