Place chicken, cider, broth, onion, mustard, vinegar, thyme, salt and pepper in a covered Dutch oven. Bring to a gentle simmer over medium heat.
Reduce heat to low and simmer covered for 20 to 25 minutes until chicken reaches 165°F, turning the breasts once. Lift chicken to a plate.
Simmer the sauce uncovered for 5 to 8 minutes until reduced slightly. Stir cornstarch with cold water, whisk into the sauce and simmer for 1 minute until it lightly coats a spoon.
Shred chicken with two forks and stir it into the sauce. Divide filling into 6 shallow containers and refrigerate within 2 hours until fully chilled.
Freeze after cooking: seal and label the chilled portions and freeze for up to 2 months. Store rolls separately, freezing them in their own bag if needed.
Thaw filling overnight in the refrigerator. Reheat in a covered saucepan over medium-low heat for 8 to 12 minutes, adding reserved water as needed and stirring, until the chicken reaches 165°F.
Warm split rolls briefly in a toaster oven and fill each with one portion of hot pulled chicken.