In a large saucepan, combine the lamb shank, onion, the halved carrot, 2 teaspoons salt and 3 quarts water. Bring to a simmer over medium-high, occasionally skimming off and discarding any foam that rises to the surface. Reduce to medium and simmer, uncovered, until a skewer inserted into the meat meets no resistance, 2 to 2½ hours.

Meanwhile, in a small saucepan, combine the oil, Aleppo pepper and garlic. Cook over medium, stirring, until the oil begins to bubble and turn red, 1 to 2 minutes. Remove the pan from the heat and cool for about 30 minutes. Strain the oil mixture through a fine-mesh sieve set over a liquid measuring cup or small bowl; discard the solids and set the oil aside.

When the lamb is done, transfer it to a large plate. Using a slotted spoon, remove and discard the carrot and onion from the broth. Add the chopped carrot and potato to the broth, bring to a simmer over medium and cook, stirring occasionally, until the vegetables are tender, 12 to 15 minutes. Meanwhile, shred the lamb into bite-sized pieces, discarding the bone and any gristle. Remove the pan from the heat.

In a medium bowl, whisk together the flour, yogurt, lemon juice and egg until well combined. Remove a ladleful of the hot broth from the pan and, while whisking constantly, drizzle it into the flour-yogurt mixture to temper it. Repeat with another ladleful of hot broth. Whisk the mixture into the broth, then stir in the shredded meat. Bring to a gentle simmer over medium, stirring constantly, until heated through and lightly thickened, about 2 minutes; do not allow the soup to boil. Remove the pan from the heat.

Stir in half each of the dill, mint and parsley. Taste and season with salt and black pepper. Serve the soup sprinkled with the remaining herbs and drizzled with the infused oil.