Heat expeller-pressed canola oil in medium saucepan. Add onion and cook over medium heat until it begins to turn transparent. Add mushrooms and cook until edges of mushrooms are browned.
Add carrots, garlic and barley and cook, mixing for 3 – 5 more minutes.
Add thyme, bay leaves and 2 cups chicken broth. Bring to a boil. Reduce heat, cover and simmer until broth is almost absorbed, about 5 minutes. Add remaining broth 1/2 cup at a time; as it gets absorbed, add more until barley is tender, about 45 – 50 minutes.