Place a large pot on medium-high heat. Add the olive oil, followed by the onion and cook until it turns golden, about 8 minutes. Stir in the garlic, fennel or celery, and ½ of the grated carrot. Cook for 5 minutes, stirring occasionally to ensure it doesn't stick to the pan.
Add the chicken breast and place a lid on the pan. Poach until the juices run clear in the thickest part of the breast, about 15 minutes, or longer as needed. I prefer to cook the chicken in the IP, under pressure for 15 minutes (use the soup or broth setting).
Remove the chicken from the soup and set aside to cool. Shred the chicken. Use two forks to shred the chicken and set aside.
Add the barley to the pan. Stir until the barley is fully incorporated. Add the bay leaves and chicken stock. Bring to a boil, then lower the heat to allow the soup to simmer.
Add the juice from half a lemon into the soup. Stir and cover with the lid slightly ajar. Leave to simmer for a further 30 minutes or until the pearl barley is tender. Check and stir the soup occasionally.
Take an immersion blender place it into one side of the pot. Blend just a little of the soup to thicken. If you don’t have an immersion blender, ladle a few cups of the soup into a standard blender, blend, and then mix it back into the soup.
Stir in the yogurt, shredded chicken breast, remaining grated carrot, and cilantro (save a little cilantro for garnish). Season to taste with salt, pepper, and more lemon juice, if you like.
Simmer the soup for a few minutes to ensure the shredded chicken is warmed through. Put in a serving bowl. Finish with a sprinkle of chopped cilantro and serve warm.