Preheat the oven to 425°. In a small bowl, mix the salt, smoked paprika, and granu- lated garlic until fully combined. Set aside.
Place the carrots cut side down on a large, rimmed baking pan. Drizzle 2 tablespoons of the olive oil over them, and toss until evenly coated. Sprinkle 1 teaspoon of the prepared spice mixture evenly across the carrots. Spread them out flat and even again in the pan, and cook for 15 minutes. Remove the pan from the oven. Push the carrots to one end of the pan to make room for the asparagus.
Toss the asparagus with 1 tablespoon of olive oil, and then spread the spears out evenly across the cleared half of the baking pan. Sprinkle 1 teaspoon of the spice mixture evenly across the asparagus and the carrots. Return the baking pan to the oven, and cook for 5 minutes. Remove the pan from the oven, and set the vegetables aside on a platter.
Add the chicken drumsticks to the empty sheet pan. Sprinkle enough of the spice mixture over the drumsticks to cover on all sides, and then spread them out evenly. Place the pan in the oven, and cook for 15 minutes. Turn the pieces over, and then cook for another 15 minutes. Add the carrots and asparagus to the pan, arranging them in an even layer around the drumsticks. Cook for 5 minutes, or until the chicken has reached an internal temperature of at least 165°. Remove the pan from the oven, and set it aside while you prepare the gastrique. (A gastrique infuses vinegar and a sweetener of some form, in this case honey, with flavoring agents of your choice. Its purpose is to bring out the full flavor and saltiness of meats.)
Place the honey in a saucepan over medium heat. Whisk constantly for 2 minutes until foamy. Add the vinegar and stir constantly for 3 minutes, and then remove the pan from the heat. Let the mixture sit for 2 minutes, and then stir in the mustard.
Drizzle the gastrique evenly over the chicken and vegetables. Sprinkle 1 tablespoon of chopped tarragon evenly over the pan, toss gently to coat both sides, and then spread the ingredients back out. Sprinkle the remaining tablespoon of chopped tarragon evenly across the entire pan, and then serve immediately.