Prepare the filling mixture by combining cream cheese and mustard in a bowl.
Spread the cream cheese mixture evenly over each tortilla.
Layer the ham slices on top of the cream cheese mixture.
Sprinkle the shredded cheddar cheese and chopped chives over the ham.
Roll each tortilla tightly from one end to another.
Wrap the rolled tortillas in plastic wrap and refrigerate for at least 30 minutes.
Remove the wraps and slice the pinwheels into bite-sized pieces.
Serve immediately or store for later use with your favorite dipping sauce.