Heat oil in a large stockpot or Dutch oven over medium heat.
Add onion and celery and cook, stirring, until beginning to soften, 2 to 4 minutes.
Grate or finely chop 1 clove garlic; add to the pan and cook, stirring, until fragrant, about 1 minute.
Add broth, chicken and barley. Bring to a gentle simmer. Cover and cook over low heat until the chicken is cooked through, about 20 minutes.
Transfer the chicken to a plate with a slotted spoon. Return the broth to a simmer and cook until the barley is tender, 20 to 30 minutes. Meanwhile, shred the chicken or cut into bite-size pieces; discard the bone.
When the barley is done, add the chicken, tomatoes and juice, asparagus, peas, salt and a grinding of pepper; return to a simmer. Cover and cook over low heat until the asparagus is tender, about 5 minutes more.
Notes
This recipe is 4 Green, 2 Blue, and 1 Purple WW points.