In a small bowl, combine Dijon mustard, shallots, garlic, thyme, salt, and pepper.
Preparing The Chicken:
Place chicken drumsticks in a large dish and liberally rub the mustard paste over the drumsticks. Cover, refrigerate, and let rest for 1-2 hours.
Roasting The Chicken:
Preheat the oven to 450 degrees F. Cover baking sheet with foil and place the mustard paste-coated drumsticks on the foil. Sprinkle the drumsticks with a little kosher salt and freshly ground black pepper and drizzle a total of 1 tablespoon olive oil over top. Roast for 20 minutes. Remove the baking sheet from the oven and drizzle an additional 1 tablespoon olive oil over the drumsticks. Turn the oven down to 350 degrees F and put the baking sheet back into the oven, rotating the pan. Roast for an additional 10 minutes, then baste the drumsticks with the juices on the foil. Roast for another 10-20 minutes, or until the outside of the chicken is deep golden brown. The internal temperature of the chicken should be 170 to 175 degrees F when a thermometer is inserted into the thickest part of one of the drumsticks and the juices run clear. Let the chicken rest for about 10 minutes before serving.