Heat olive oil in a large saucepan over medium-high heat. Once hot, add the shallot and ¼ teaspoon of salt, and sauté until soft and translucent, about 3 to 4 minutes.
Add garlic, cabbage, barley, and the zest from 1 lemon and sauté for an additional 3 to 4 minutes. Once the cabbage is soft and tender, stir in the chicken stock, Parmesan Meatballs, and the remaining ¾ teaspoon salt. Reduce the heat to low and cook for 13 to 15 minutes.
In a small bowl, combine cornstarch and 1 tablespoon of water, mixing until smooth. Stir the cornstarch slurry into the soup and cook until slightly thickened.
Cut the zested lemon into 4 wedges. Set aside.
Stir in the dill, parsley, and the juice from 2 lemon wedges. Continue cooking for 2 to 3 minutes. Taste and season with additional salt, as needed.
Divide the soup between 2 bowls and garnish each with 1 to 2 tablespoons of feta crumbles, dill, parsley leaves, freshly ground black pepper, and a lemon wedge for squeezing.