Pat the chicken drumsticks dry and season them evenly with sea salt and black pepper.
Heat the olive oil in a small heavy-bottomed pot or Dutch oven over medium-high heat.
Sear the drumsticks until golden brown on all sides, then remove them from the pot and set aside.
Add the diced onion and carrot to the pot, sautéing for 3-4 minutes until the onion is translucent.
Stir in the minced garlic and dried thyme, cooking for another 30 seconds until aromatic.
Pour in the white wine to deglaze the pan, scraping up any browned bits from the bottom.
Add the chicken bone broth and return the drumsticks to the pot, ensuring they are partially submerged.
Cover and reduce heat to low, simmering for 25-30 minutes until the chicken is tender and cooked through.
While the chicken braises, prepare the white rice according to package instructions.
Serve the braised chicken and vegetables over the fluffy rice, spooning the remaining pan juices over the top.