Step 1: Preheat your oven to 400°F (200°C) and line a large baking sheet with parchment paper or a silicone mat for easy cleanup.
Step 2: In a medium saucepan, combine the apple cider, apple cider vinegar, and brown sugar. Bring to a boil over medium-high heat, then reduce heat to medium and simmer until the liquid has reduced to about 1/2 cup (20-25 minutes).
Step 3: Remove the reduction from heat and whisk in the Dijon mustard, minced garlic, chopped rosemary, thyme, and smoked paprika to complete the glaze for your apple cider glazed chicken drumsticks.
Step 4: Pat the chicken drumsticks dry with paper towels. Place them in a large bowl, drizzle with olive oil, and season generously with salt and pepper. Toss to coat evenly.
Step 5: Arrange the chicken on the prepared baking sheet. Bake for 25 minutes. Then, remove from the oven and brush a thick layer of the glaze onto each of the apple cider glazed chicken drumsticks.
Step 6: Return the chicken to the oven and bake for another 15-20 minutes, brushing with more glaze every 5-7 minutes, until the internal temperature reaches 165°F and the glaze is sticky and caramelized.
Step 7: Let the apple cider glazed chicken drumsticks rest for 5 minutes before serving. Drizzle with any remaining glaze from the pan.