Preheat a large pot or Dutch oven with 2 tablespoons of EVOO over medium-high heat.
Season the drumsticks with salt and freshly ground black pepper. Place flour in a shallow dish and coat the drumsticks, shaking off any excess. Add half of the coated chicken to the hot oil and brown on all sides. Once the chicken is browned all over, remove from the pot to a plate to reserve. Add another 2 tablespoons of EVOO to the pot and brown the second batch of drumsticks. Remove them to the plate with the first batch.