In a large bowl, combine the yogurt, lemon juice, ginger, garlic, garam masala, cumin, turmeric, Kashmiri chili powder, and 3/4 teaspoon salt. Add the chicken and stir until every piece is coated. Cover and refrigerate for 30 minutes.

While the chicken marinates, place the rinsed basmati rice in a bowl and cover with cool water. Soak for 20 minutes, then drain thoroughly in a fine-mesh strainer.

Stir the saffron threads into the 2 tablespoons (30 ml) warm milk. Set aside while preparing the remaining components. The milk should gradually turn golden.

Fill a 4- to 5-quart saucepan with plenty of water. Add 1 tablespoon salt, the cinnamon stick, cardamom pods, cloves, and bay leaf. Bring to a rolling boil over high heat.

Add the soaked and drained rice: Boil uncovered for about 5 to 7 minutes, stirring gently once or twice. The grains should be roughly 70% cooked: flexible on the outside but still slightly firm in the center.

Drain immediately through a fine-mesh strainer: Remove and discard the cinnamon stick, cardamom pods, cloves, and bay leaf when convenient. Do not rinse the cooked rice.

Heat 1 tablespoon (14 g) butter and the neutral oil in a 5- to 6-quart Dutch oven over medium heat. Add the diced onion and cook for 5 to 7 minutes, stirring occasionally, until soft and lightly golden.

Stir in the ginger and garlic and cook for about 30 seconds. Add the garam masala, cumin, coriander, and Kashmiri chili powder. Cook for another 30 seconds, stirring constantly, until fragrant.

Add the tomato puree and tomato paste: Cook over medium heat for 4 to 5 minutes, stirring frequently, until the sauce thickens slightly and becomes deeper red.

Add the chicken along with all of its marinade. Cook for 7 to 9 minutes, stirring occasionally, until the pieces are mostly cooked and the sauce looks thick rather than watery.

Reduce the heat to low: Stir in the heavy cream, remaining 2 tablespoons (28 g) butter, and 1/2 teaspoon salt. Simmer gently for 2 to 3 minutes. The sauce should coat the chicken without leaving a large pool of thin liquid.

The chicken should be nearly or fully cooked before layering. If checking with an instant-read thermometer, the largest pieces should reach 165°F (74°C) by the end of the final steaming stage.

Keep the chicken and sauce spread evenly over the bottom of the Dutch oven. Spoon half of the parboiled rice gently over the chicken without stirring the layers together.

Sprinkle half of the cilantro, half of the mint, and half of the 1/2 cup (30 g) crispy fried onions over the rice.

Add the remaining rice in an even layer: Sprinkle with the remaining cilantro, mint, and fried onions.

Drizzle the saffron milk over several areas of the rice rather than mixing it throughout. Dot the top with the remaining 1 tablespoon (14 g) butter.

Cover the Dutch oven tightly: Cook over the lowest possible heat for 18 to 20 minutes. Avoid opening the lid during this stage because trapped steam finishes cooking the rice.

Remove the lid carefully: The rice grains should be tender, long, and separate. Check a large chicken piece from the center; it must reach 165°F (74°C).

Remove the pot from the heat, replace the lid, and let the biryani rest for 10 minutes. This allows excess steam to settle and helps the rice firm slightly.

Starting at one side of the pot, gently lift portions from the bottom upward so each serving contains both butter chicken and fragrant rice. Avoid vigorous stirring, which can break the grains. Garnish with additional fried onions, cilantro, mint, or lemon wedges if desired.