Season chicken breasts generously with salt and pepper; place in a large skillet and add 1/2 inch of water. Simmer, covered, over medium-low heat until chicken is opaque throughout, 15 to 20 minutes; remove chicken from skillet. When cool enough to handle, shred chicken with fingers, discarding skin and bones; chill until room temperature.
In a large bowl, whisk together sour cream, mayonnaise, chutney, and mustard.
Add chicken, scallions, and raisins; season with salt and pepper and stir to combine.