For the spice mix: Combine the spices in a small bowl and set beside the stove to use throughout cooking.
For the rice mixture: Bring 2 tablespoons water and 2 tablespoons oil to a simmer in a large pan. Sprinkle potato cubes with ½ teaspoon spice mix and ½ teaspoon salt. Add to simmering water along with 1 tablespoon lemon juice. After 5 minutes, cover and turn heat to medium low. Cook, stirring occasionally, until the potatoes are fork tender, about 15 minutes. Empty cooked potatoes into a large bowl and set aside.
Bring ½ tablespoon water and ½ tablespoon oil to a simmer in a small pan. Add a pinch of spice mix along with raisins. Sauté for 5–10 minutes until raisins have plumped up. Add to the bowl of potatoes.
Add a splash of oil to the small pan along with a pinch of spice mix. Add almonds and fry for 2 minutes. Add to the bowl of potatoes.
Add ½ tablespoon oil to the small pan along with a splash of water and the dried cherries. Add a pinch of spice mix and stir over high heat for 5–10 minutes until cherries plump up and change color. Add to the bowl of potatoes.
To make the meatballs: Combine ground lamb with chopped onions, tomato paste, pomegranate molasses, ½ teaspoon spice mix, ¼ teaspoon salt, and ¼ teaspoon pepper. Mix with your hands and form into 1-inch balls. Heat ¼ cup oil in a large pan and fry the meatballs until they are cooked through and browned, about 10 minutes. Add to the bowl of potatoes.
To parboil the rice: Fill a large pot with 4 cups water plus 1 tablespoon oil, ½ teaspoon salt, and 1 tablespoon lemon juice. Bring to a boil as you start preparing the lamb and dried apricot sauce.
For the lamb and dried apricot sauce: Heat 1 tablespoon oil in a medium skillet. Add and sear lamb stew meat with the chopped onion. Stir in ½ tablespoon spice mix and sauté for 1–2 minutes. Add enough water to cover the meat by 1–2 inches. Cover and simmer for about 30 minutes until tender. Check occasionally, adding more water as needed. Add dried apricots plus enough water to cover fruit and meat by 1–2 inches. Cover pan and cook another 30 minutes, adding more water as needed.
When the rice water has come to a boil, drain the soaked rice and add to the boiling water. Cook (uncovered) for 10–15 minutes. (The water will not be completely absorbed.) Drain rice through a large sieve or colander, reserving ½ cup cooking water. Run cold water over the draining rice to cool it.
Oil the bottom and sides of a large pot with a tight-fitting lid. Drape 2–3 tortillas to cover the bottom and sides of the pot. Spread a layer of rice over the tortillas and add about ⅓ of the potato mixture, gently mixing it into the rice. Sprinkle with some spice mix. Repeat these steps two more times, ending with rice on top.
Place the reserved ½ cup of rice cooking water in the small pan, add butter and heat to melt, then pour over the layered rice. Cover rice with tortillas and cover the pot with a tight-fitting lid.
Turn heat to medium high until steam comes up through the rice. Turn the heat down to medium-low for 10–15 minutes until rice is done. [Our test kitchen preferred completing this step in the oven at 350° for 30 minutes instead of on the stovetop.]
Remove the pot lid and place a large serving platter over the pot. Invert to unmold the dish. Serve with the lamb and dried apricot sauce.