Zest the lemons and limes. Chop zest strands fine. Juice the fruit, discarding the rind and interior pith. Combine lemon and lime juices, mustards, rice vinegar and olive oil and whisk until blended. Remove the leaves from the stems of tarragon.
Chop leaves fine.
Chop chives fine. Add chopped herbs to the emulsified dressing and season to taste with pepper and salt. Store dressing tightly sealed in refrigerator for 1-2 weeks. When the water reaches a rapid boil, add the spaghetti and cook until tender, about 9 minutes. Meanwhile, if the chicken is frozen, run it under hot water so you can remove any packaging.