Kinda like a peanut butter sandwich (a very lean one) ...
The main thing to avoid is having any squirt out when you clamp the cooler on.
The short answer is "NO", you do not apply the gunk to both sides of the sandwich. ... I use a flat utility razor and (with a syringe) I "draw" a thin, squarish, doughnut/bead about half way between the center and the edge of the CPU chip ... then, I use the razor blade (the flat-back variety) to coax and wipe the bead into a smooth layer, about like a single coat of interior latex, so that it covers the whole middle, but does not come closer than 20% (~25%) to the edge of the "pin package" (the CPU "proper" only occupies < the center centimeterof the larger pin/wafer substrate). ...
ALSO! ... Some cooling plates come with a "convenient" (tacky) greyish coating of thermal conductive "paste" which looks like teflon, or like a lotto scratch-off ....
... I remove (carefully razor-scrape) this "grattis gunk" from the cooler before I apply paste to the CPU package. Do one, or the other, but not both! If you know what you are doing, or if you really are going to over-clock, then scrape the "EZ cote" and lay down your own (juicier) coating of quality thermal compound.
... Again ... "too much" is the greater crime ... messy, at best.
I have coached several of my Pals, on this (directly) and, I always set up two bright lights, on a large clean desk/table (no shadows) ... a clean, old tee-shirt and a few razors, business card, paper towel (not fuzzy). Magnifying glass ... etc.
Also a good setup for thos crappy case manuals and recessed system jumpers and headers.
Inserting the CPU (into the socket) and thermal paste are the only truly touchy operations, for the first few builds. Perfection is just fine. No rush? Right?
You can carefully wipe off any excess and start over, if need be .. just do not let that goo get into the socket or the socket gap, traces, pins, etc.
=g'luck=