Pat the chicken drumsticks completely dry. Coat with olive oil and season with Cajun seasoning, smoked paprika, garlic powder, onion powder, thyme, black pepper, and salt as needed.
Heat a large deep skillet or Dutch oven over medium-high heat. Sear the drumsticks for 3 to 4 minutes per side until browned. Transfer to a plate.
Reduce the heat to medium. Add the vegetable oil and flour. Cook for 3 to 4 minutes, stirring continuously, until the roux reaches a medium caramel-brown color.
Add the onion, bell pepper, and celery. Cook for 5 to 7 minutes. Add garlic and cook for 30 seconds.
Gradually pour in the chicken broth while whisking. Add Worcestershire sauce, Cajun seasoning, smoked paprika, thyme, and optional cayenne.
Bring the gravy to a gentle simmer and cook for 3 to 5 minutes.
Return the drumsticks to the skillet and spoon gravy and vegetables over the chicken.
Cover, reduce the heat to medium-low, and simmer for 30 to 35 minutes, turning once.
Check the thickest part of the chicken without touching the bone. It must reach at least 165°F.
Remove the lid and simmer for 5 to 10 minutes if the gravy needs to thicken.
Stir in the butter, taste, and adjust the seasoning.
Serve the chicken and gravy over white rice. Finish with parsley or green onions.