Place the chicken in a deep, large roasting pan. Using your fingers, gently tug the skin from the meat over the breasts and drumsticks. Rub the butter between the skin and the meat. Rub 2 teaspoons salt and 2 teaspoons sumac all over the chicken skin. If time permits, refrigerate the chicken, uncovered, for at least 6 hours and up to overnight.
Preheat the oven to 425°F (220°̊C).
Squeeze half of 1 lemon over the chicken and place it inside the cavity along with a half head of garlic. Drizzle the chicken with 2 to 4 tablespoons oil.
In a large bowl, combine the carrots, potatoes, onions, and the remaining lemons and garlic. Pour over 1/4 cup oil, sprinkle with 1 teaspoon sumac, and season with salt and pepper. Toss until the vegetables are evenly coated and scatter them around the chicken.
Roast, tossing the vegetables occasionally, until the chicken and vegetables are deep golden and cooked through and the chicken registers 165°F (72°C) in the thickest part of the thigh, 60 to 75 minutes. Allow the chicken to rest for 10 minutes before carving.
Squeeze the garlic out of its papery husk and stir it into the vegetables.
Drizzle the labneh with oil and sprinkle it with the sesame seeds, some pepper, sumac, and flaky sea salt. Serve the labneh alongside the chicken.