Preheat the oven to 425°F (232°C).
Clean and pat the drumsticks dry with a paper towel. Place them a bowl. Add oil (2 tbsp), salt, pepper, garlic, thyme and rub the seasoning on the drumsticks. Place the cut potatoes, carrots, onions into a 13" x 9" baking pan/dish. Add 2 tbsp oil, season with salt & pepper. Mix everything together.
Place the seasoned drumsticks on top of the vegetables. Pour the unsalted chicken stock in the baking pan/dish (slowly pour the broth near the edges of the dish and not over the chicken).
Place the dish in the oven and bake for 25 minutes. Remove the pan/dish from the oven and flip the drumsticks. Return the dish into the oven again and bake for another 35 minutes. Remove the pan/dish from the oven. Transfer the drumsticks to a plate (cover with foil) and set aside.
In a small bowl: combine cornstarch and cold water, stir until the cornstarch has dissolved. Pour the mix in the pan/dish. Gently mix by swirling the vegetables around to evenly distribute the conrstarch mix (the gravy will thicken within several minutes).