Put chicken, cider, broth, onion, mustard, vinegar, thyme, salt, and pepper in covered Dutch oven. Gently simmer over medium.
Reduce to low; simmer covered 20 to 25 minutes to 165°F, turning breasts once. Plate chicken.
Simmer sauce uncovered 5 to 8 minutes to reduce slightly. Mix cornstarch and cold water; whisk in and simmer 1 minute until lightly spoon-coating.
Shred chicken with two forks; stir into sauce. Divide among 6 shallow containers; refrigerate within 2 hours until fully chilled.
Seal and label chilled portions; freeze up to 2 months. Store rolls separately, freezing in their own bag if needed.
Thaw filling overnight in refrigerator. Reheat covered in saucepan over medium-low 8 to 12 minutes to 165°F, stirring and adding reserved water as needed.
Briefly warm split rolls in toaster oven; fill each with one portion hot chicken.