Preheat the oven to 425 degrees F.
Season the chicken generously with salt and pepper. Heat the oil and 1 tablespoon butter in a Dutch oven (12-inch diameter) over medium-high heat. Lay half of the chicken pieces skin-side down in the skillet. Cook the chicken, until golden-brown on both sides, turning once, about 8 minutes. Transfer to a plate. Repeat with remaining chicken.
Reduce the heat to medium, add the carrots and celery, and cook until just tender, about 10 minutes. Add the rosemary, and cook until fragrant, about 1 minute more. Add the lemon juice, and scrape up any browned bits that cling to the pan with a wooden spoon. Add the broth, onions, lemon zest, and 1 teaspoon salt and bring just to a boil.
Return the chicken skin-side up to the pot, taking care that the skin is just over the surface of the liquid, and then put the stew in the oven. Cook, uncovered, until the chicken is cooked through, tender, and lightly glazed and brown, about 40 minutes. Baste the chicken occasionally in the last 20 minutes of cooking. Meanwhile, use a fork to work the remaining 2 tablespoons butter into the 2 tablespoons flour to make a paste.
Remove the stew from the oven. Remove and discard the rosemary and arrange all the vegetables and chicken (reserving the juices in the pan) in a serving dish/bowl. Place the Dutch oven over medium heat and whisk the butter/flour mixture into the juices until dissolved. Bring to a boil and simmer until thickened. Pour sauce over the chicken and serve immediately over buttered noodles.