Adjust over rack to middle position and heat to 450 degrees. Place drumsticks, carrots, celery, onions, and garlic in roasting pan, spray with vegetable oil, and toss well. Roast, stirring occasionally, until well browned, about 1 ½ hours.
Transfer contents of roasting pan to Dutch oven. Add broth, wine and thyme and bring to a boil over high heat. Reduce heat to low and simmer until reduced by half, about 2 hours. Pour through fine-mesh strainer into large container (discard solids), cover with plastic wrap, and refrigerate until fat congeals, at least 2 hours.
Using soup spoon, skim fat and reserve. Heat ½ cup fat in Dutch oven over medium high heat until bubbling. Wisk in flour and cook, whisking constantly, until honey colored, about 2 minutes. Gradually whisk in stock, bring to boil, reduce heat to medium-low, and simmer until slightly thickened, about 10 minutes. Season with salt and pepper.