Heat the oven to 425°F.
Cut the lemons in half crosswise and squeeze 2 Tbs. juice from one half; set the juice aside in a dish. Reserve the squeezed half for the chicken cavity. Cut each of the 3 remaining halves in half again for a total of 6 pieces.
Coat the bottom of a small flameproof roasting pan with the oil.
Wash and pat the chicken dry. Remove any excess fat. Season the chicken inside and out with salt and pepper.
Put the squeezed lemon half inside the chicken along with the thyme, rosemary, and garlic cloves. Truss the bird by crossing the legs and tying them lightly with string and tucking the wings behind the back.
Brush the chicken all over with the melted butter. Sprinkle with herbes de Provence. Set the chicken on its side in the roasting pan, using the 6 lemon pieces, flesh side down, to support it.
Roast the chicken for 20 min., and then turn the bird on its other side, turning the lemon pieces as well to their other flesh side. Roast for another 20 min. Turn the bird on its back and turn the lemons skin side down. Reduce the oven temperature to 350°F. Add 1/4 cup of the vermouth to the pan, stir to mix with the pan juices, and then baste the chicken. Roast the chicken until an instant-read thermometer inserted into the thickest part of the thigh reads 170°F, about 40 min. more.
Transfer the chicken to a carving board, placing the chicken breast side down. Tent with aluminum foil, and let rest for 10 to 15 min.
Discard the roasted lemons. Tip the roasting pan and spoon out and discard as much fat as you can, leaving behind the juices. Set the pan over medium-high heat. Add the chicken broth and deglaze the pan by scraping up the browned bits from the bottom of the pan with a wooden spoon. Bring to a boil and reduce to 2/3 cup, about 3 min.
Mix the cornstarch and 1 Tbs. of the reserved lemon juice in a dish. Whisk this into the pan mixture. Return to a boil for about two minutes to thicken, and then remove the pan from the heat.
Remove the trussing string from the legs and carve the chicken. (Discard the herbs, lemon, and garlic from the cavity.) Add any juices from the chicken to the sauce. Taste the sauce, and adjust the seasoning with salt, pepper, and the remaining lemon juice and vermouth. The balance of flavors will depend on the acidity of the lemons and your taste. Serve the chicken with the sauce.