Place the chicken in a 12- or 14-inch cast-iron pan, breast-side up. Top with 2 tablespoons butter and generously sprinkle with salt and pepper. Roast the chicken, 5 minutes. Remove the chicken and carefully shake it to ensure that it does not stick to the bottom of the pan. Return to the oven to roast, 20 more minutes. Baste the chicken and continue to roast, basting every 5 minutes, for 30 minutes. Add the carrots, garlic and onions and continue to roast until the chicken is cooked through and an instant-read thermometer inserted in the thigh (avoiding bone) reads 155 degrees F, about 20 minutes more.
Place the chicken on a cutting board with the vegetables. Skim the fat from the pan, leaving about 2 tablespoons. Place the pan over a burner and add the flour. Cook until browned, then add the wine. Scrape out all the pan juices and the wine-flour mixture into a small saucepan. Cook for 3 minutes, then add the cream and remaining 2 tablespoons butter, whisking. Season with salt and pepper. Slice the meat off the bird and add the cooking juices to the gravy. Place the sliced meat on a platter and sauce with the gravy. Garnish with the vegetables.