In a 5-quart Dutch oven or large pot, heat 2 tablespoons oil over medium-high. Add onions and garlic and cook until caramelized and tender, stirring frequently, about 15 minutes, scraping up the browned bits (add a little water to the bottom of the pot if it begins to overbrown). Season with salt and pepper.
Add barley and 6 cups water and bring to a boil.
Reduce heat; simmer partially covered over medium
until barley is tender, about 40 minutes. Stir in beans
and chickpeas and cook until heated through, about
1 minute. Remove from heat, stir in the 2 cups whole
parsley leaves. Add more water to thin soup as necessary. Season with salt and pepper.
Meanwhile, in blender, combine dill, mint, chopped
parsley, vinegar, remaining 1/4 cup oil, and 2 tablespoons water. Puree to desired consistency. Serve soup topped with herb pesto and a dollop of yogurt.