1. Whisk all the marinade ingredients together in a medium bowl. Add the chicken drumsticks to the bowl and toss them around. Cover with plastic wrap and refrigerate for at least 30 minutes or overnight.
2. In a large sauté pan, heat the olive oil over medium to high heat. Add the chicken and cook until the meat is no longer pink, about 10 minutes, turning once to cook all sides.
3. Add half the marinade to the chicken and continue cooking until the marinade turns into a thicker sauce and the chicken is fully cooked, 20 to 30 minutes. Cover and keep warm over low heat.
4. Cook the remaining marinade in a small saucepan over medium heat for about five to 10 minutes, until it reduces about half and thickens a bit.
5. Serve the chicken over rice, drizzling with sauce.