Soak the rice for 30 minutes in lukewarm water that covers it generously. Drain. While the rice is soaking, put the oil in a medium pan and set over medium-high heat. When it is hot, add the shallots. Stir and fry for 3 to 4 minutes, or until the shallots are lightly browned. Put in the green beans and stir them around for 3 minutes. Now add the tomato paste and tomatoes as well as 3 tablespoons of water (or liquid from the can of tomatoes), 1/4 teaspoon, of the 3/4 teaspoon salt. Stir and bring to a simmer. Cover, turn the heat down to low, and simmer gently for 12 to 13 minutes, or until the beans are tender. Add the lime and stir to mix. Set aside.
Put 10 cups of water into a large pot and bring to a rolling boil. Add 1 1/2 tablespoons salt and mix in. Scatter the rice into the boiling water and bring to a boil again. Boil rapidly for 51/2 minutes, or until the rice is almost done but still has a very slim hard core in its very center. Drain the rice immediately and leave in a sieve or colander.
Put the butter in a nonstick pan and set it over low heat. Add 2 tablespoons of water and turmeric. Peel the potato and cut it crosswise into 1/8-inch-thick rounds. Lay a round in the center of the pan. Surround it with the other rounds until you have no more space left. You can cut some of the rounds if needed to almost cover the bottom. A few blank spaces are fine. As soon as all the butter has melted, divide the rice into 3 parts and spread 1 part over the potatoes. Top this rice with half of the green beans, spreading them out evenly. Sprinkle 1/4 teaspoon cinnamon over them. Cover the beans with the final layer of rice. Cover the pot and turn the heat to medium-high. After 4 minutes, turn the heat down to medium-low. After another 4 minutes, lift the cover and quickly drape a dish towel over its underside and then put it back on the pan. (The towel will be between the pan and the lid.) Flip the ends of the towel on top of the lid so they do not burn. Turn the heat down to very low and cook 30 minutes. To serve: Have a large, warm serving plate ready. Remove the cover and slide a knife along the inside of the pan to loosen the rice. Put the serving plate on top of the rice. Now upturn it and invert its contents, rice and crust, onto your serving plate. The "cake'' will crumble a little, spreading out at the bottom, but this is as it should be.