Preheat the oven to 375 degrees. Season drumsticks with salt, pepper and 1 teaspoon of the smoked paprika and place on a shallow sheet pan. Place in the middle of the oven and allow to cook for 40 minutes.
Place the remaining ingredients in a saucepan on medium heat. Stir to combine.
When the sauce starts to boil reduce heat and simmer 20 minutes more. Stirring occasionally.
Transfer sauce mixture to the bowl of a blender or food processor and process until smooth. Transfer to a bowl or storage container.
During the last 20 minutes of cooking, brush one side of the drumsticks with some of the sauce and return to the oven for 10 minutes. Flip the drumsticks and brush the other side with the sauce, return to the oven and allow to cook for 10 more minutes.
Remove the chicken to a serving plate, and serve it up with a side of extra barbecue sauce.