Bring a Stockpot to a boil and season generously with salt. Add the chicken breasts and simmer for 20–25 minutes, or until a thermometer inserted into the thickest part registers 165°F. Remove from the water and set aside.
Add the Greek yogurt, mayonnaise, Dijon mustard, lemon zest, lemon juice, tarragon, chives, kosher salt, and black pepper to a bowl fitted with the paddle attachment.
Break the warm chicken into large pieces and add to the mixer. Mix on low speed to begin shredding the chicken, then increase to medium-low speed and mix until the desired consistency is reached, about 1 minute.
If not using a mixer, whisk together the yogurt mixture in a Mixing Bowl. Shred the warm chicken with two forks and toss with the dressing until well combined.
For meal prep, pack a generous scoop of chicken salad with 5–6 Bibb lettuce leaves, cornichons, and crackers into the On the Go Lunch Box. Refrigerate until ready to serve.
To serve, fill the lettuce leaves with chicken salad and top with sliced tomatoes and cornichons, or enjoy the chicken salad spooned onto crackers.