Put the rice in a bowl and fill with cold water. Swish the rice with your fingers to release excess starch, then pour off the water. Repeat 2 or 3 more times until the water is less cloudy. Drain and set aside.
Pour the milk into a heavy-duty 5- to 6-quart pot and bring to a boil over mediumhigh heat, stirring occasionally. Meanwhile, tie the cinnamon sticks and cardamom pods in a piece of cheesecloth.
When the milk begins to boil, lower the heat to maintain a gentle simmer and add the spice pouch. Cook, stirring frequently, until the milk has reduced to 5 cups, about 45 minutes.
Add the rice and a pinch of salt and continue to cook, stirring gently and frequently, until the rice is very soft (but not mushy) and the pudding has started to thicken, 25 to 40 minutes. Remove the spice pouch.
Add the jaggery and stir until dissolved. Season to taste with more jaggery and salt, if necessary. Remove from the heat and let stand, covered, at room temperature for 30 minutes to thicken; it should be thick enough to pour slowly from a spoon. If a skin forms, stir it back into the pudding. Serve warm, garnished with the toasted coconut, if using.