Is it truly impossible to tell what a CPU is doing?
Are there techniques an electrical engineer could use to verify that a circuit actually performs the operations described in its spec, and no other operations?

In theory, yes, I think this is possible. However, for a complex CPU it will take a lot of time and money. Also, if you do not fully know and understand the design, you will be unable to judge if any activity is "legit" or not.

A CPU is "just" a complex digital circuit consisting of many logic cells.

It is possible to reverse engineer the chip and reconstruct the design by observing the metal connections. There can be many of these connection layers like up to 8 layers or more.

You will need experts in the field to recognize the logic cells and then maybe some software can figure out how they're all connected so you can reconstruct the netlist.

Once you have the netlist you "know" the design. That doesn't mean you now also know how it works!

It could be that a certain function activates 2 sections of the design while you think one should be enough so you then suspect some suspicious activity is going on. However, the design does some clever trick you do not know about to speed up operations.

Without knowing and understanding the design, any conclusion you draw might still be wrong. Only the engineers who designed the CPU have all the design information and stand the best chance of being able to figure out or guess what actually goes on or should go on in a CPU.