If using dried apricots, at least 40 minutes before cooking, cut them into quarters, soak in warm water to cover, then drain and set aside. (This can be done up to a day ahead. Cover and refrigerate, then bring to room temperature before using.)
Warm the oil in a medium saucepan set over medium-high heat. Add the onion, season with ½ teaspoon of salt and cook, stirring often, until the onion begins to turn golden at the edges, about 5 minutes. Stir in the tomato and cook for 5 minutes more, stirring frequently.
Pour 5 cups of the vegetable broth into the pot, scraping any stuck-on bits from the bottom, and bring to a boil. Add the lentils, cover, and reduce the heat to maintain a strong simmer. Cook, stirring once or twice, until the lentils are soft, 15 to 20 minutes.
Stir in 1 more cup of broth if the soup is too thick, then add the carrots, apricots and ½ teaspoon salt, and season with pepper to taste. Increase the heat to bring to a boil again, then reduce the heat and simmer, partly covered and stirring once or twice, for 10 minutes, or until the carrots are as cooked as you’d like.
Turn off the heat and stir in the lemon juice, then taste. If it’s not sour enough, add a little more, and adjust for salt and pepper.
Ladle into bowls and serve warm, with pomegranate syrup drizzled over the top.