Using a 4 to 5-quart Dutch oven on medium-high heat, add 2 tablespoons oil.
Add the chicken drumsticks, and sauté for 3 minutes on each side. Remove the drumsticks and set them aside.
Add the remaining oil to the Dutch oven. Add the onion, and sauté uncovered for 2 minutes, stirring often. Keep the heat on medium-high.
Add the garlic and ginger. Continue cooking and stirring often, for 1 to 1½ minutes. Keep the heat on medium-high.
Next add the 2 cans of Coca-Cola, soy sauce, brown sugar, black pepper, and salt. Continue stirring often, and cook uncovered for 4 to 5 minutes.
Reduce the heat to medium, and add back in the chicken drumsticks. Cook for 20 minutes uncovered, turning the chicken once.
After cooking the chicken for 20 minutes, remove the chicken from the pot and set aside. Increase the heat to medium-high. Allow the sauce to reduce for 5 to 8 additional minutes. Once the sauce reduces, remove the pot from the heat. Ladle the sauce over the chicken, and serve while hot.