In a large mixing bowl, combine the shredded chicken, softened cream cheese, buffalo wing sauce, shredded cheese, green onions, and finely diced celery until well blended.
Optional: Slightly warm the flour tortillas to make them more pliable and prevent cracking.
Lay each tortilla flat and spread the chicken mixture evenly across the surface, leaving a small margin at the edges.
Roll each tortilla up tightly to form a log.
Wrap each log tightly in plastic wrap to help it maintain its shape.
Refrigerate the wrapped rolls for at least 30 minutes, though 2-4 hours is recommended for the neatest slices.
Remove from the refrigerator and unwrap. Using a sharp serrated knife, slice the logs into 1-inch thick pinwheels.
If necessary, secure each pinwheel with a toothpick.
Arrange on a platter and serve chilled with ranch or blue cheese dressing and fresh vegetable sticks.