How to Make It
Soak the rice in a large bowl of lukewarm water for 30 minutes. Drain well.
Meanwhile, heat the oil in a medium saucepan until shimmering. Add the shallots and cook over moderately high heat, stirring, until lightly browned, about 4 minutes. Add the beans and cook, stirring, until bright green but still crisp, about 4 minutes. Add the tomatoes, tomato paste, 1/4 teaspoon of the cinnamon, 3/4 teaspoon of salt and 3 tablespoons of water and bring to a simmer. Cover and cook over low heat until the beans are tender, about 12 minutes. Stir in the lime juice and transfer to a bowl.
In a large pot, combine 10 cups of water with 1 1/2 tablespoons of salt and bring to a boil. Stir in the rice and return to a boil. Cook the rice until barely tender, about 5 minutes; drain.
Melt the butter in a 3-quart nonstick saucepan over low heat. Stir in the remaining 2 tablespoons of water and the turmeric. Arrange the potato slices, slightly overlapping, in the bottom of the saucepan. Spread one-third of the rice over the potatoes and top with half of the green bean mixture and 1/4 teaspoon of the cinnamon. Repeat the layering, ending with a final layer of rice.
Pat the rice into a compact cake, cover and cook over moderately high heat for 4 minutes. Reduce the heat to moderately low and cook for 4 minutes. Remove the lid and stretch a kitchen towel over the saucepan, being sure not to let the towel touch the rice. Cover with the lid and fold the corners of the towel up and away from the heat. Cook the rice over very low heat for 30 minutes.
Remove the lid and towel and run a thin bladed spatula around the edge of the saucepan. Set a large plate over the saucepan and carefully invert the pilaf onto the plate. Cut the pilaf into wedges and serve immediately.