In a large bowl, mix chicken and sauce. Slice rolls in half horizontally and pull centers from tops and bottoms. Divide chicken mixture evenly among bottoms; set tops in place. Pack in a cooler to transport (sandwiches can be chilled up to 4 hours).
Balsamic barbecue sauce: In a 2-quart pan, combine 1 cup ketchup, 3/4 cup balsamic vinegar, 1/3 cup firmly packed brown sugar, 1/4 cup molasses, 1 tablespoon Worcestershire, 1 minced garlic clove, 1 teaspoon dry mustard, 1 teaspoon ground ginger, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Simmer, uncovered, over medium-low heat, stirring occasionally, until mixture is reduced to about 2 cups, 12 to 15 minutes. Use warm or at room temperature. Makes 2 cups.