I removed the handles and the top four front drawer plates.
At the hardware store, I got them to cut the four pieces of plywood to the right size. Then, I sandpapered and varnished the plywood (twice).
My brother-in-law helped me with drilling the right size holes according to the original drawer front. He also made holes for the handles. (I gave him one drawer front to use as a template).
He advised me not to route the groove where the bottom of the drawer fits in the drawer front. Rather, to cut a small piece of the front of the drawer, put the front together and screws a small corner profile on the bottom so it stays perfectly fixed.
KNARREVIK with plywood topper
We still had some plywood left over, and since we just moved into a new home, we did not want to invest that much in every room already.
But I did want to have a pair of matching nightstands for a low price. The cheapest prices nightstands often are made of metal. I thought the sound of metal could wake somebody up (if you put your phone/watch on your nightstand for example).