Preheat the oven to 180°C/fan 160°C/gas mark 4. Remove the skin from the chicken drumsticks.
Heat the olive oil in a large, flameproof casserole dish and fry the drumsticks until browned. Scatter the onions and garlic over the drumsticks, and cook until browned.
Stir the stock and vegetable soup mix into the casserole dish and season. Bring to a simmer, then cover and transfer to the oven and bake for 45 minutes. Leave to stand for 5 minutes before serving to allow the casserole mixture to thicken slightly.
Meanwhile, steam or boil the broccoli florets for 5 minutes, drain and serve with the casserole.