Heat the olive oil in a large Dutch oven over medium-high heat. Season the chicken drumsticks evenly with the sea salt and freshly cracked pepper, to taste. Place into the hot Dutch oven and cook, until golden brown, about 3 minutes.
Flip the chicken drumsticks over and place the uncovered Dutch oven into the preheated oven. Cook for 25-30 minutes, or until chicken is cooked through. Remove from the oven and place the chicken on a plate then cover loosely with a tin foil tent.
Place the Dutch oven on the stove over medium-high heat. Add the apple cider and Dijon mustard then whisk together making sure to scrape up the browned bits on the bottom of the pan. Whisk, while it boils, until it reduces down a bit and thickens up, about 2-3 minutes. Taste and season with sea salt and freshly cracked pepper, if needed.
Place the drumsticks and their juices back into the Dutch oven and toss to coat the chicken evenly. Place on a platter then serve with additional sauce on the side. Enjoy.