I have had no issues with simple cutouts, but when a cutter is like a butterfly, for instance, the little tail breaks off at times. Or If its an apple, the stem may break off.
I do chill the rolled out dough (between parchment) before I cut out and take great care to slightly assist the thin areas as I lift the cutout.
What I notice is that the dough may have some layers at times even though I have rolled it.
Please help and thank you!