Season chicken breasts 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.