How to Create a Perforated Piezoelectric Disc (aka atomizer)?
This has all the markings of an Engineering SE question.
According to a number of websites, American Piezo included, the commercially produced product is composed of various materials, but include a ceramic/crystalline component.

A traditional piezoelectric ceramic is a mass of perovskite ceramic
crystals, each consisting of a small, tetravalent metal ion, usually
titanium or zirconium, in a lattice of larger, divalent metal ions,
usually lead or barium, and O2- ions.

One can expect the ceramic/crystalline component is what contributes to the brittle characteristic of the material. Traditional cutting tools are likely to create as much damage as the attempt to puncture created.
A carefully controlled water jet may be able to create a hole, but the size is not likely to be as small as you desire. According to Accurate Water Jet, you can get holes in 3 mm material as small as 500 microns but the diameter is not going to be consistent. The process involves forcing high pressure water with abrasive directly at the surface, which results in spray back along with a sideways vector, abrading the upper portion of the hole. The size is outside of your requirements, which makes this an exercise in futility.
Abrasion applied by a rotating object may be a solution. I've read of glass being cut by copper tubing. The tubing is placed in a drill press, a dam of modeling clay constructed around the desired area and water with abrasive poured into the dam. Light pressure and repeated lifting allows fresh abrasive along with cooling. It's not fast, but broken glass is akin to broken ceramic piezo disks.
About the best DIY method you can try would be similar. Buy the tiny bits, but grind the tip flat. Build a dam, load it with jewelers rouge in water suspension and dab away. If you are using an ordinary drill press, consider that the run-out for such a machine may exceed the diameter of the bit. This will result in a larger hole and/or destruction of the bit.
How to determine amount of pressure? Use my favorite method for tightening a screw: apply torque until it snaps, then back off a quarter of a turn. Seriously, too little force will get you farther than too much, for obvious reasons. You aren't trying to force the bit through the ceramic. You're trying to apply the rouge to the disk while it spins.
High speed means higher heat, so a moderate rpm is indicated, along with the light pressure.
In response to OP comment:
I own a CO2 laser and for some reason neglected to consider that as an option. I can create a hole in plywood approximately 0.110 mm, which is still larger than your requirement. I can create similar holes in glass which has properties similar to ceramic and vice versa. Returning from a test moments ago, I've discovered the laser will create coning in a manner similar to that expected of a water jet. The laser creates microscopic fracturing caused by the heat of the laser being absorbed by the glass. In a pane of ordinary window glass, the upper surface was approximately 1.50 mm in diameter when the laser reached the lower surface. The hole in the lower surface was 0.50 mm in diameter.
Diode lasers, which have a lower cost than CO2 lasers will not impact glass due to the different frequency of the beam. The focused beam of a diode laser is rarely a circle or square shape and would not provide a suitable hole either way.