Preheat the oven to 425°F and line a large baking sheet with parchment paper.

Prepare the saffron milk by bringing the nondairy milk to a boil in a small saucepan, then stir in the saffron and remove from the heat. Stir in the rose water and set aside to infuse.

Rinse the rice in a fine-mesh sieve under cool water, swishing the grains until the water runs clear, then drain. Bring a large pot of water to a boil with the black cardamom, green cardamom, mace, bay leaf, cinnamon stick, cloves, and salt. Add the rice and simmer until the grains are about three-quarters cooked, 5 to 8 minutes—they should still have a slight bite. Drain well, keeping the whole spices with the rice.

Toss the squash with oil and salt on the prepared baking sheet, spreading the pieces into an even layer so they are not touching. Roast until the edges are starting to brown, about 30 minutes, then sprinkle with garam masala and adjust salt to taste.

Sear the sliced mushrooms in a wide pan over medium-high heat with oil until they begin to brown, about 5 minutes. Stir in the garam masala and salt, then transfer to a bowl and reduce the oven temperature to 375°F.

Build the masala in the same pan by heating oil and toasting the cumin and fennel seeds until they start to brown and smell nutty, 1 to 2 minutes. Lower the heat to medium, add the sliced fennel, onions, and salt, and cook until the vegetables are softened and deeply caramelized, about 20 minutes. Stir in the garlic and ginger until fragrant.

Add the coriander, ground cumin, garam masala, turmeric, and red chili, toasting the spices for about 30 seconds until aromatic. Remove the pan from the heat and stir in the yogurt, roasted squash, seared mushrooms, cilantro, and mint, seasoning generously with salt and extra chili if desired.

Layer the biryani in a 12-inch round baking dish or Dutch oven, starting with one-third of the parboiled rice and spices, followed by a tablespoon of saffron milk and two-thirds of the vegetable mixture. Continue layering with another third of rice, more saffron milk, the remaining vegetables, and finally the last of the rice and saffron milk.

Cover the dish tightly—use foil under the lid if the seal is loose to trap the steam—and bake for 30 minutes until the rice is tender and the flavors have melded. Garnish with fennel fronds, extra herbs, and cashews, and serve hot.