As John Grant pointed out, the Cricut is one of the worst choices
for cutting relatively thick paper.  Laser cutting is a good choice
but most people do not have access to a suitable laser cutting machine.
The really cheap Chinese machines, in the one watt range, are barely
capable of surface scorching, not cutting.  There are many maker
spaces across the globe and many have good laser cutting machines.

There are many ways to make paper or cork gaskets.  The best method
is mainly determined by the quantity you need.  If you need just a few
a sharp scalpel is all you need.  A few more and it is worth making
a template or printing the pattern on paper to be cut out.  If you
need a larger quantity and the cuts are simple, such as circles,
a few tubular punches can be used with or without an alignment fixture
or by making a "steel rule die" by conventional methods.

I made a simple shuttling jig for cutting any combination of IDs and
ODs for items such as valve faces and small rings.  I lent it to a
friend since I no longer do player work.  He has used it successfully
for all the punchings needed for his Wurlitzer pipe organ.  Pace
punches sells these tubular punches at reasonable prices.

If you are going into production and need to make a lot of different
sizes, it makes sense to buy one of the fancy French concentric ring
systems.  If you need to make large quantities of one particular size
it might be worth paying to have a one-piece die made.

Die making is much cheaper than it used to be due to methods such as
EDM machining.  It is no longer necessary to laboriously machine finish
and harden a simple die.  You cut the pattern in reverse in soft carbon
and use this to burn the die right into hardened steel.  A few years
ago I paid something like $150 for a punch and die set that was much
more complex than a simple ring cutter.  Since I own a very precise CNC
router with a high speed spindle I would probably experiment with this
the next time I need a thick gasket.

Spencer Chase