Sprinkle lamb all over with ½ teaspoon salt. In a large pot, combine lamb, onion, bay leaf, 6 cups of water (1.4 liters), and bring to a gentle boil over medium-high heat. Cover and cook, reducing heat as needed to maintain a gentle simmer, until the meat is fully tender and falls off the bone easily, 1 to 1 ½ hours. Using a slotted spoon, transfer the meat to a cutting board and let cool slightly, about 15 minutes. Strain the meat stock through a fine-mesh strainer set over a large bowl; Set aside.

Once meat is cool enough to handle, use 2 forks or your hands to debone and shred the meat into thin long strands; set aside.

While the lamb cooks, in a large Dutch oven or large heavy-bottomed stockpot, combine the cracked wheat, 8 cups of water (1.9 liters), and remaining 1 teaspoon salt and bring to a gentle boil over medium-high heat. Reduce heat to maintain a gentle simmer, partially cover the pot, and cook, stirring occasionally to prevent sticking and scorching on the bottom of the pot. Cook until the wheat is softened and mixture is very thick and creamy, about 1 ½ hours, adjusting with additional lamb stock or water as needed to keep grains fully submerged to reach proper consistency.

Ladle half of the cooked wheat into a food processor and process until smooth, about 1 minute.

Pour the processed wheat mixture into a large fine-mesh strainer set over a large mixing bowl. Using a silicone spatula or a flexible dough scraper, push the mixture through the strainer until only the tough pieces of cooked cracked wheat remain. Scrape and clear the underside of the strainer as needed. This process may take up to 10 minutes. Discard the unpassable tough wheat. Repeat processing and straining the remaining cooked wheat.

When done, you should have between 1.5 to 2 quarts (1.4 to 1.9 liters) of a velvety, light cream color, soft paste that has the consistency of mayonnaise. This is referred to as the 'syrup of wheat.'

Rinse and dry the now-empty heavy-bottomed stock pot that you had cooked the wheat in and transfer the 'syrup of wheat' back into the clean pot.

Set ¼ cup of the shredded lamb aside, then stir the remaining shredded lamb into the cooked wheat in stockpot. Add a ½ cup of the reserved meat stock into the mixture. With a wooden spoon or a large silicone spatula, stir to combine. Cook over medium-low heat and while continually stirring, slowly pour in additional meat stock, ½ cup at time, until the mixture is a thick creamy consistency, about 15 minutes. Lower the heat as needed so the mixture is steaming hot but not boiling. Add butter and stir until butter is fully melted, about 2 minutes.

While continuing to cook the porridge over low heat, use a handheld mixer on its lowest speed to whip the porridge until the haleem is velvety, stretchy, and elastic with the strands of meat broken down into fine pieces, 10 to 15 minutes.

Transfer the cooked haleem to a large serving bowl or individual bowls and garnish with the reserved shredded meat. Serve with your preferred optional garnishes.