Finely dice onion. Heat butter or oil in a medium-sized soup pot over medium heat. Add diced onion and sweat, stirring occasionally, until soft and translucent, 5-7 minutes. Do not brown during this time. If onion starts to brown, turn down heat. Add dried mint and cook for 1 minute.
Add broth and bring to a boil, then add barley and salt. Cover and cook for 45 minutes, until barley is cooked through. Turn off heat.
Put yogurt in a 4 cup liquid measuring cup or bowl. Add some warm broth from the soup into the yogurt to temper it. Stir until evenly mixed, and add the yogurt mixture slowly back to the soup. Ladle into bowls and serve with fresh mint leaves.