Rinse the basmati rice in cold water until the water runs clear, then soak for 30 minutes.
In a bowl, combine the yogurt, biryani masala, ginger garlic paste, salt, and chicken drumsticks. Let it marinate for at least 1 hour.
Bring the water to a boil in a large pot. Add the soaked rice and cook for 5 to 6 minutes until 70% cooked, then drain well.
Heat the ghee in a heavy-bottomed Dutch oven over medium heat. Fry the sliced onions until golden brown, then remove half for later use.
Add the marinated chicken to the pot with the remaining onions and cook for 15 minutes until the chicken is lightly browned and tender.
Layer the parboiled rice evenly over the cooked chicken. Top with the reserved fried onions, chopped cilantro, and fresh mint.
Cover the pot tightly with a lid and cook on very low heat using the dum method for 15 to 20 minutes.
Remove from heat and let the biryani rest for 10 minutes. Gently fluff the rice with a fork and serve hot with lemon wedges.