Tender-crisp peas add a bright bite to this comfort food staple. Serve with warm biscuits.
Bring 6 1/2 cups broth to a simmer in a medium pot over high heat. Add chicken, cover, and reduce heat to maintain a very gentle simmer. Poach until chicken is cooked through, 25 to 30 minutes. Transfer chicken to a plate; let cool slightly.
Add green onions and carrots to broth, increase heat to medium, and simmer until green onions are just tender, 4 to 5 minutes. Add pasta and simmer until just tender, 6 to 8 minutes. Meanwhile, shred chicken from bones.
Stir chicken, cream, snap peas, parsley, salt, and pepper into soup. Remove from heat and let stand a couple of minutes until snap peas are bright green. Add more broth if you like. Stir in tarragon and ladle into bowls.