Preheat oven to 350 degrees F.
Combine olive oil, brown sugar, paprika, thyme, oregano, garlic powder, 1 teaspoon kosher salt, and black pepper in a small bowl, and mix into a thin paste. Spoon this mixture over the drumsticks and gently rub in to coat. Set aside.
Melt butter in a medium saucepan over medium-high heat. Add onion and cook for 3-4 minutes, just until softened. Add garlic and cook for 30-60 seconds, just until fragrant, then scrape the onion and garlic mixture out into an 8”x10” or 9”x13” baking dish. Add rice and remaining 1/2 teaspoon salt to the baking dish and stir well.
Add broth and water to the same saucepan you used for the onion. Bring these to a boil, then pour the hot liquid directly over the rice and stir it in.
Arrange drumsticks on top of the rice, along with about half of the lemon wedges. Cover the baking dish tightly with foil. Bake for 30 minutes, then remove the foil and bake for another 20 minutes until the rice is tender and the chicken is fully cooked.
Remove from the oven and let rest for 2-3 minutes. Remove the drumsticks and lemons to a plate, then use a fork to gently fluff up the rice. Garnish with extra lemon wedges and a few sprigs of parsley or thyme if desired, and serve.