Warm the vegetable oil in a large heavy pot over medium heat, then add the finely chopped onion and cook, stirring often, until deep golden, 10–12 minutes.
Stir in the turmeric and minced garlic and cook until fragrant, about 1 minute.
Add the rinsed yellow split peas and the water, bring to a boil, skim any foam, then reduce to a gentle simmer and cook until the peas are just tender, 30–35 minutes.
Add the rinsed rice, the finely chopped parsley, cilantro, and mint, and simmer until the rice is tender and the soup begins to thicken, 12–15 minutes.
Meanwhile, combine the ground lamb with the grated and squeezed onion and the beaten egg; mix just until sticky, chill 10 minutes, then roll into 1-inch meatballs (about 24).
Brown the meatballs in 1 tbsp of the remaining oil in a skillet over medium heat until lightly colored, 3–4 minutes, then slide them into the simmering soup.
Stir in the unsweetened pomegranate juice and pomegranate molasses; return to a gentle simmer and cook until the meatballs are cooked through and the peas are soft, 20–25 minutes. Balance the tartness by stirring in the granulated sugar.
Season the soup with the salt and ground black pepper; adjust to taste and thin with hot water if needed for a thick-but-sippable consistency.
For the mint oil, warm the remaining oil in a small skillet over low heat until shimmering, remove from heat, and immediately stir in the dried mint; let bloom 30 seconds.
Ladle the soup into bowls, drizzle with the mint oil, and scatter with pomegranate seeds (for serving). Serve hot.