Heat the olive oil in a medium Dutch oven over medium heat. Add the carrots, celery and leek and cook, stirring occasionally, until the leek softens, 3 to 4 minutes. Add the cabbage, paprika, 1/2 teaspoon salt and a few grinds of pepper and stir to coat. Increase the heat to medium-high and cook, stirring often, until the cabbage is wilted and the vegetables start browning, 4 to 5 minutes.

Add 2 cups water, the chicken broth and chopped chicken to the pot; bring to a simmer. Add the barley, cover and simmer until tender, about 10 minutes; season with salt and pepper.

Divide the soup among bowls. Top with the sour cream, dill and more paprika.