Let the chicken sit at room temperature 30 minutes. Meanwhile, preheat the oven to 425˚. Remove 4 strips zest from the lemon with a vegetable peeler. Cut the lemon in half and set aside. Toss the sliced fennel, garlic, onion and lemon zest with 1 tablespoon vegetable oil in a roasting pan; set a rack on top. Place the chicken breast-side up on the rack. Squeeze 1/2 lemon over the chicken, then rub with the remaining 1 tablespoon vegetable oil; season inside and out with 2 teaspoons salt and 1/2 teaspoon pepper, then place the squeezed lemon half and fennel tops (breaking to fit) in the cavity. Tie the legs together with kitchen twine.
Transfer the chicken to the oven and roast 45 minutes. Mix the broth and vermouth in a measuring cup. Spoon a few tablespoons over the chicken and into the pan. Continue roasting, basting every 15 minutes with the broth mixture, until the juices run clear and a thermometer inserted into the thickest part of the thigh (not touching the bone) registers 165˚, 45 minutes to 1 hour 15 minutes (halfway through, tip the cavity juices into the pan). You should have about 1/2 cup of the broth mixture left.
Let the chicken rest 10 minutes in the pan. Tip the cavity juices into the pan again, then transfer the chicken to a cutting board; let rest 30 minutes. Meanwhile, remove the rack from the pan and pour in any remaining broth mixture, scraping up any browned bits; gently smash the vegetables. Pour the broth and vegetables into a fine-mesh sieve set over a medium saucepan; press to release the juices (discard the solids).
Bring the juices to a simmer over medium heat; season with salt and pepper. Remove from the heat, add the butter and swirl to melt. Add 1 teaspoon lemon juice from the remaining 1/2 lemon. Carve the chicken and serve with the gravy.