In a bowl or ziplock bag, combine the chicken, oregano, garlic, olive oil, and the juice from half a lemon. Season with salt and pepper, then mix well. Marinate in the fridge for at least 30 minutes, or overnight if you prefer.
Preheat the oven to 350°F. Arrange the marinated chicken drumsticks in a single layer in a square baking dish, discarding any leftover marinade. Bake for 15 minutes.
OPTIONAL STEP: In a skillet, heat the butter over medium heat and sauté the onion until softened. Set aside.
Remove the chicken from the baking dish and set it on a plate. Add the chicken stock, rice, juice from the remaining half of the lemon, and the sautéed onion (if using) to the baking dish. Stir to combine. Nestle the chicken drumsticks back on top of the rice mixture.
Return the dish to the oven and bake for an additional 25-30 minutes, or until the rice has absorbed all the liquid and the chicken is fully cooked.
Let the dish stand uncovered for a few minutes. Fluff the rice with a fork and serve with Greek tzatziki.