Place the olive oil in a large stock pot. Set the pot over medium heat and add the onions. Sauté the onions for 2-3 minutes, then add the garlic and cook another 2 minutes. Stir so the garlic doesn't burn.
Place whole chicken breasts, barley, thyme, saffron, and 1/2 teaspoon salt in the pot. Pour in the chicken broth and bring to a boil. Allow the soup to simmer for 20 minutes, stirring occasionally. Then remove the chicken breasts with tongs.
While the soup is simmering, stir in the cream cheese and kale. Chop the chicken breasts and stir the chicken pieces back into the soup. Taste, then salt and pepper as needed.