2 to 3 pounds chicken drumsticks

For the Brine:
2 quarts cold water
½ cup kosher salt
¼ cup light brown sugar

For the Smoked Paprika Rub:
3 tablespoons smoked paprika
3 tablespoons salt
4 teaspoons onion powder
2 teaspoons garlic powder
2 teaspoons ground white pepper
1½ teaspoons freshly ground black pepper
¼ teaspoon dried thyme
¼ teaspoon dried oregano

For the Spicy Apple Glaze:
3 cups apple cider
1 cup cider vinegar
1 cup applesauce
1 cup granulated sugar
½ cup light brown sugar
¼ cup thinly sliced fresh ginger
½ cinnamon stick (about 1½ inches)
½ vanilla bean, split, seeds scraped out and reserved
6 black cardamom pods
2 teaspoons minced garlic
2 teaspoons allspice berries
1½ teaspoons crushed red pepper
½ teaspoon dry mustard
¼ teaspoon salt