Heat oven to 375 degrees F. Spray large cookie sheet with cooking spray.
If using crescent rolls: Unroll dough; press into 12x8-inch rectangle, firmly pressing perforations to seal. If using dough sheet: Unroll dough; press into 12x8-inch rectangle.
Spread cranberry chutney on top of rectangle, within 1/4 inch of edges. Arrange sliced turkey evenly over cranberry chutney.
Sprinkle shredded cheese over sliced turkey.
Starting with longest side, roll up; pinch long side to seal. With serrated knife, cut into 24 slices.