Melt 1 tablespoon butter in a large skillet over medium heat. Add onion, and cook 10 minutes or until soft and translucent. Add chicken and next 7 ingredients. Set aside.
Beat broth and egg with a wire whisk until well blended; pour into a small skillet. Cook over medium heat 2 to 3 minutes, stirring constantly, until soft scrambled. Fold egg mixture into chicken mixture.
Lay 1 sheet of phyllo dough on a flat surface; brush evenly with melted butter. Repeat procedure with 3 more phyllo sheets. Stack sheets, and cut in half lengthwise to form two phyllo stacks. Spoon 1/3 cup filling down long edge of each phyllo stack, spreading evenly to ends. Roll up, starting with long side, and brush edges with more butter to seal.
Cut roll into 4 pieces; transfer pieces to an ungreased baking sheet. Repeat with remaining phyllo sheets, butter, and filling.
Bake at 350° for 15 to 20 minutes or until golden brown.