Add 1 tablespoon of oil to a large skillet, then sear cubed lamb in batches. Once browned, add to a six-quart Dutch oven and top with 6 cups of water. Simmer, covered, for 45 minutes to an hour or until tender.
Slice eggplant into ¼" slices and sprinkle with salt on both sides. Place on paper towel-lined tray and allow to sweat for 15 minutes.
Spray a foil-lined baking sheet with cooking spray, then add sliced eggplant in a single layer. Spray the tops generously with cooking spray, then bake for 30 minutes in a 375℉ oven. Flip half-way through.
While the meat and eggplant are cooking, slice potatoes into ¼" slices and sprinkle with salt on both sides. Fry in the same pan used to sear the meat. If necessary add 1 tablespoon of oil, or more as needed. Once potatoes are golden brown, remove from pan and set aside until needed.
Finally, slice onion in half, then into sections and slice peppers into rings. Sauté both in the same skillet, over low heat, for approximately 10 minutes until the onion is caramelized.
Once the meat is done cooking, strain the broth into a bowl and clean the pot. Pour 4 cups of broth into a large measuring cup and stir in 2 tablespoons butter, 1 tablespoon salt, and ½ can (3 ounces) of tomato paste. If you want to add biryani seasoning, do so now. Whisk until the tomato paste is completely dissolved.
Wash rice thoroughly to remove starch, then strain and add to the cleaned pot. Pour broth-tomato sauce mixture over it and stir. Bring to a boil, then cover and cook over low heat for 15 to 20 minutes. Stir once or twice to cook evenly.
Spray bottom and sides of an 8-quart Dutch oven with nonstick cooking spray. Place a tortilla on the bottom, then arrange eggplant, potatoes, onions, peppers, and meat in layers over the tortilla.
Spoon cooked rice into the pot and add ½-1 cup of broth. Press down on the rice to compress it as tightly as possible. Cover and cook over very low heat for an additional 15 minutes.
Rest Makloubeh for 15 minutes, then remove the lid and cover the pot with a rimmed flat platter or tray, larger than the pot itself. While pressing the tray tightly onto the pot, flip the pot quickly to un-mold. Carefully peel away the tortilla before serving.