Combine the rice and water in a medium bowl and soak for 20 minutes. Drain and reserve the water. Set the rice and water aside separately.
Meanwhile, toast the saffron in a small skillet over medium heat, stirring and pulling the skillet off the heat occasionally to keep the saffron from burning, until the saffron darkens to a maroon color and is fragrant, 15 to 20 seconds. Transfer the threads of saffron to a mortar and crush to a coarse powder with a pestle (alternately, you can crush the saffron in a bowl with the back of a spoon). Stir in the milk or cream and set aside.
Combine the ghee or oil, cinnamon stick, cardamom, and ginger in a medium heavy-bottomed casserole over medium-high heat. Cook, stirring, until the cinnamon unfurls, 1 to 2 minutes.
Add the currants and nuts and cook, stirring, for 1 minute. Add the drained rice and cook, stirring, for 1 minute. Add the reserved water, turn down the heat to low, cover, and simmer very gently for 15 minutes.
Uncover the rice and sprinkle it evenly with sugar and drizzle the saffron mixture over the top. Reduce the heat to very, very low heat, cover, and continue to cook for 5 more minutes. Serve hot.