Place chicken breasts in a medium saucepan with broth and bay leaf. Bring to a gentle simmer over medium heat, then reduce heat to low, cover, and cook until chicken reaches 165°F internally, 15-18 minutes. Remove from heat and let chicken rest in hot liquid for 5 minutes. Transfer to a plate and cool completely, then shred into bite-sized pieces with two forks.

In a food processor, combine parsley, basil, chives, anchovies, and garlic. Pulse until finely chopped, scraping down sides as needed, about 30 seconds. Add lemon juice and Dijon mustard and pulse to combine.

Add mayonnaise and sour cream to the processor. Blend until smooth and bright green, about 1 minute. Taste and season with salt and pepper - the anchovies add salt, so start light. The dressing should be thick but pourable; thin with a teaspoon of water if needed.

In a large bowl, toss shredded chicken with celery and scallions. Add about three-quarters of the dressing and fold gently until evenly coated. Taste and add more dressing as desired, or reserve extra for serving. Season with additional salt and pepper if needed. Chill for at least 30 minutes before serving to let flavors meld.