Bring lentils, 4 cups water, and 1 teaspoon salt to boil in medium saucepan over high heat. Reduce heat to low and cook until lentils are tender, 15 to 17 minutes. Drain and set aside. While lentils cook, place rice in medium bowl and cover by 2 inches with hot tap water, let stand for 15 minutes.
Using your hands, gently swish rice grains to release excess starch. Carefully pour off water, leaving rice in bowl. Add cold tap water to rice and pour off water. Repeat adding and pouring off cold tap water 4 or 5 times, until water runs almost clear. Drain rice in fine-mesh strainer.
Heat reserved 3 tablespoons onion oil, garlic, coriander, cumin, cinnamon, allspice, cayenne, and 1/4 teaspoon pepper in Dutch oven over medium heat until fragrant, about 2 minutes. Add rice and cook, stirring occasionally, until edges of rice begin to turn translucent, about 3 minutes. Add 2 1/4 cups water, sugar, and 1 teaspoon salt and bring to boil. Stir in lentils and figs, reduce heat to low, cover, and cook until all liquid is absorbed, about 12 minutes.
Off heat, remove lid, fold dish towel in half, and place over pot; replace lid. Let stand for 10 minutes. Fluff rice and lentils with fork and stir in cilantro and half of crispy onions. Transfer to serving platter, top with remaining crispy onions, and serve, passing yogurt sauce separately.