Special equipment: A small roasting pan and rack, butcher's twine
Pat the chicken dry inside and out with paper towels, and let sit at room temperature to take the chill off, about 30 minutes. Meanwhile, position an oven rack in the center of the oven, and preheat to 450 degrees F. Put a rack in a small roasting pan.
Season the cavity of the chicken generously with salt and pepper, and stuff it with the thyme and garlic. Tie the legs together with butcher's twine. Season the outside with salt (generously) and a few grinds of pepper, and place the chicken breast-side up on the rack. (Seasoning the chicken when on the rack could result in gravy that's too salty.)
Roast the chicken until the skin is golden brown and crispy and a thermometer inserted in the deepest part of the thigh registers 165 to 170 degrees F, depending on how you like your dark meat, 50 minutes to 1 hour. Let the chicken rest on the rack for a few minutes, then carefully tip it forward to drain all the collected juices into the roasting pan. Discard the thyme and garlic. Let the chicken rest on a cutting board while you make the gravy. (To transfer the chicken easily, hold a wooden spoon handle in the cavity with one hand and use the other hand to lift the chicken with tongs.)
Remove the rack from the pan, and pour the pan drippings into a small bowl. Skim off 2 tablespoons of the fat, and stir them into the flour and 2 tablespoons water in another small bowl. Heat the roasting pan over medium-high heat, add the wine, scrape up any browned bits and simmer until the wine reduces by half, 3 to 4 minutes. Add the chicken stock and any remaining pan drippings, bring to a simmer, add the flour mixture and whisk constantly until the liquid thickens into a gravy, 2 to 3 minutes. Season the gravy with salt and pepper.
Remove the butcher's twine from the chicken. Snip off the wingtips using kitchen shears; reserve to make stock. Remove the leg quarters by slicing between the thigh and the body, then finding the hip joints near the backbone and slicing through them. Separate the drumsticks and thighs, and transfer to a platter. Sit the carcass upright; snip out the backbone and reserve it to make stock. Lay the breast skin-side up and slice in half through the breastbone. Separate the wings and breast, and transfer to the platter. Serve the chicken with the warm gravy. Reserve the carcass to make stock.
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