Pat the chicken drumsticks dry with paper towels and season lightly on all sides with salt and freshly cracked black pepper. Set aside.
Heat 1 Tbsp olive oil in a large skillet over medium-high heat. Add the chicken drumsticks and sear until golden brown on all sides, about 2–3 minutes per side. The chicken should not be cooked through. Transfer to a plate and set aside.
In the same skillet, add another 1 Tbsp olive oil over medium heat. Add the garlic and ginger and sauté for 30 seconds until softened and fragrant. Add in the red chilis (or chili flakes) and chopped scallion and cook for another 30 seconds.
Add the diced onion and mushrooms and cook for 1–2 minutes until slightly softened. Pour in the coconut milk and chicken broth, then add fish sauce, sugar, lime zest, lime juice, and salt. Stir well and bring to a gentle simmer.
Return the chicken to the pan, nestling it into the coconut lime broth. Reduce heat to low, cover, and simmer for 15-20 minutes, turning once halfway through, until the chicken is fully cooked, tender, and slightly falling off the bone. Taste the broth and adjust seasoning as needed.
Serve the chicken over steamed white rice and a side of vegetables, spooning plenty of coconut lime broth on top. Finish with chopped scallions and a drizzle of chili oil.
