Chain shortening only one link? Is it possible to shorten or extend a chain only one link as shown in the photo (in red square). Or do I have to shorten at least two links in order for the other end to match?

The reason I'm asking is that with my current setup (front 52/36, rear 32/11) if I remove one link the chain becomes too short to shift to 52 front and 32 rear. If I extend the chain one link and shift to 36 front and 11 rear, the chain rubs on the derailleur.
For 1/8" chains as used on singlespeed bikes (or with hub gears), half links are reasonably common.

They do exist for 3/32" chains (which are sometimes used without derailleurs) but are much less common as single speed 3/32" chains are. They're not needed if you've got a test derailleur as the derailleur provides much more chain length adjustment. A chain tensioner can do this on a singlespeed.

The main reason for using these appears to be on singlespeed conversions, when they lack the horizontal dropouts of a frame built for single speed.