Set a large soup pot on the stove over medium heat. Add the olive oil and let it warm for about 30 seconds.

Add the chopped onion, sliced carrots, sliced celery, and sliced cabbage to the pot. Sprinkle in 1 teaspoon of the salt and stir well.

Cook the vegetables over medium heat for 8 minutes, stirring a few times, until the onion looks soft and the cabbage has started to wilt.

Add the chopped garlic and caraway seeds to the pot. Stir for 1 minute, just until the garlic smells fragrant.

Pour the chicken broth into the pot. Turn the heat to high and bring the broth to a boil.

Stir the pearl barley into the boiling broth so it does not clump at the bottom of the pot.

Lower the heat to medium-low so the soup bubbles gently. Partly cover the pot and cook for 25 minutes.

Stir the soup and check the barley. Keep simmering for 5 to 10 minutes more, until the barley is tender with a little chew left and the vegetables are very soft.

Add the shredded rotisserie chicken to the pot. Stir to spread the chicken through the soup.

Simmer the soup for 5 minutes, stirring once or twice, until the chicken is hot all the way through.

Stir in the lemon juice, black pepper, and half of the chopped dill. Taste the broth and add the remaining 1/2 teaspoon salt if needed.

Ladle the soup into bowls. Sprinkle the remaining dill over the top and serve hot.