Preheat the oven to 375°F (190°C).
Place the turkey breast on a roasting pan. Rub olive oil evenly over the skin.
In a small bowl, combine salt, black pepper, dried thyme, dried rosemary, and minced garlic. Rub this mixture all over the turkey breast.
Roast the turkey breast in the preheated oven for about 1 hour and 15 minutes, or until the internal temperature reaches 165°F (74°C). Baste occasionally with pan juices.
While the turkey is roasting, prepare the vegetables. Toss carrots and green beans with olive oil, salt, and black pepper on a baking sheet.
After the turkey has been roasting for 45 minutes, place the vegetables in the oven alongside the turkey. Roast until tender, about 45 minutes.
Meanwhile, place the peeled and cut potatoes in a large pot and cover with cold water. Bring to a boil and cook until tender, about 15 minutes.
Drain the potatoes and return them to the pot. Add sour cream, whole milk, salt, and black pepper. Mash until smooth and creamy. Keep warm.
To make the gravy, melt butter in a saucepan over medium heat. Whisk in flour and cook for 1-2 minutes until lightly golden.
Gradually whisk in chicken broth and whole milk. Continue whisking until the gravy thickens, about 5-7 minutes. Season with salt and black pepper to taste. Keep warm.
Once the turkey is done, let it rest for 10 minutes before slicing.
Serve sliced turkey breast with mashed potatoes, roasted carrots, roasted green beans, and gravy on the side.