Mix ground ginger, garlic powder, yellow chili powder, and 1/2 teaspoon garam masala in a bowl with enough water to make a paste. Transfer the paste and yogurt to a large resealable plastic bag; add chicken, coat with yogurt marinade, squeeze out excess air, and seal bag. Marinate in refrigerator for at least 1 hour.
Bring rice, chicken stock, and white wine to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until the rice is tender and the liquid has been absorbed, 20 to 25 minutes.
Preheat oven to 350 degrees F (175 degrees C).
Keep 1/4 cup melted butter, milk, and saffron warm in a small saucepan on low heat.
Melt remaining 5 tablespoons butter in a large skillet over medium heat; cook and stir carrot, bell pepper, onion, and 2 teaspoons garam masala until vegetables are golden brown, about 10 to 15 minutes. Remove chicken from the marinade, shake off excess, and add to vegetables. Reserve remaining marinade. Cook and stir until liquid evaporates. Stir marinade, mint, ground coriander, and cilantro into chicken mixture; reduce heat to low and simmer until chicken is tender, about 10 minutes.
Spread about 1/3 of the rice in a large baking dish. Layer about 1/3 of the chicken and vegetable mixture over rice and top with 1/3 of the saffron butter mixture. Repeat the layers twice more.
Bake in preheated oven until bubbly, about 15 minutes.