Is it possible to have peltier plates within a few micron thickness range?
You can do it but there are two problems - one theoretical and one practical.

The practical problem is in bonding the two conductive plates to the TE junctions. This is currently done by soldering, and soldering two plates with microns between them is a recipe for disaster when you get solder squeezout which shorts the micron gap.

The theoretical is addressed here http://www.tec-microsystems.com/EN/Intro_Thermoelectric_Coolers.html


  Low height pellets or/and larger pellets cross-section provide higher
  cooling capacity value for thermoelectric cooler. In the same time
  they increase the operating current and total power consumption. Small
  pellets cross-section and tall pellets increase maximum temperature
  difference and reduce TEC power consumption, but cooling capacity is
  reduced too.


As for a process that will provide significant cooling over extremely small gaps, I'm not aware of any, but I'm not an expert. If nothing else, very short thermal paths by their nature have low thermal impedance, so it's a challenge to keep the temperature differential at desired levels, as the heat seeks to redistribute itself across the gradient.