Soak dried shiitakes in boiling water for 20 minutes. Strain through a fine-mesh sieve and reserve the liquid. Squeeze the mushrooms to remove excess liquid, then dice.
Meanwhile, heat oil in a large pot over medium heat. Add leeks and garlic; cook, stirring, until very soft, about 4 minutes. Add carrots, fresh shiitakes, white mushrooms, salt and pepper; cook, stirring, until the vegetables start to soften, 3 to 4 minutes.
Add chicken and cook, stirring, until it is no longer pink on the outside, about 4 minutes.
Add broth, barley, thyme, the reserved soaking liquid and chopped soaked mushrooms. Increase heat to high and bring to a simmer. Reduce heat to medium-low, partially cover and simmer until the barley is tender, about 50 minutes. Stir in vinegar.