Place the rice in a large bowl and add cold water to cover by 3 inches. Swirl the rice around with your fingers until the water becomes cloudy, then put the rice in a strainer to drain. Repeat the process until the water remains clear. This will take 4 to 6 rinsings.

Place 8 cups of water and the salt in a large shallow pot (preferably nonstick; the pot should be 10 to 12 inches across and about 6 inches deep) and bring to a boil over high heat. Reduce the heat to medium-high, add the rice, and cook, uncovered, at a brisk simmer for 6 minutes. Pour the rice into a strainer, rinse it under cool running water, and drain well. Wipe out and dry the rice pot with paper towels.

Meanwhile, place the cranberries, sugar, and 1½ cups of water in a medium-size saucepan. Bring to a simmer over medium heat and cook until the cranberries are soft and most of the cooking liquid has evaporated, 5 to 8 minutes; the juice of the cranberries should be thick and syrupy. Using a slotted spoon, transfer the cranberries to a bowl. Add the 2 tablespoons of butter to the cranberry juice and bring to a boil, stirring until the butter melts. Remove the cranberry juice from the heat and set it aside, covered, at room temperature.

Combine the yogurt, melted butter, and 2 tablespoons of water in a bowl and whisk to mix. Pour the yogurt mixture evenly over the bottom of the rice pot. Spoon a ½-inch layer of rice evenly over the yogurt mixture. Stir the cranberries into the remaining rice, and without disturbing the rice layer in the pot, add the cranberry rice, mounding it toward the center. Place the pot, uncovered, over medium heat and cook until the rice begins to brown on the bottom, 6 to 8 minutes.

Reduce the heat to low. Wrap a clean kitchen towel around the pot lid, piling any excess cloth on top of the lid. (You want to keep the cloth away from the heat.) Place the cloth-covered lid over the rice and cook until the rice on top is tender and the rice on the bottom of the pot has formed a dark golden crust, about 40 minutes. Check after 30 minutes; if you don’t see a crust, increase the heat slightly.

Remove the pot from the heat and pour the reserved cranberry juice over the rice. Let the rice stand, covered, for 5 minutes.

To serve, spoon the loose rice from the top into a bowl, leaving the crusty rice at the bottom. Place a round platter over the pot. Invert the pot and give it a little shake. The crusty rice should slide out in a golden brown disk. Cut the crust into wedges and serve these with the loose rice.