How does a flame sensor or thermocouple operate with just one wire?

The Flame rod in your picture is not a thermocouple.

The other connection in the circuit is the flame being in contact with a metal surface that acts as a ground.

Most every flame rod system I have worked on puts and AC voltage on the flame rod. Depending on the system this voltage may be between 80 and 300 volts.

Without a flame present in the system you can only verify an AC voltage from the Flame Rod to ground.

When a flame contacts the flame rod part of the flame MUST be in contact with a metal surface - such as a nozzle where the gas and air combine.

A DC electrical current is established that flows from the flame rod through the flame and to ground. The DC current can be measured with a micro amp meter in series.

I believe that the general principal is that the AC voltage ionizes carbon particles in the flame which conduct the current. Gas and air mixtures affect the flame rod signal.

IF the flame pulls away from the nozzle, such as a higher firing rate, there will be intermittent flame detection as the ground path is usually lost.

The flame sensor in your photo uses the electrode in the flame vs. the frame of the furnace (which includes the root of the flame) and is connected to the mounting lug on the flame sensor (that's why there is a ceramic insulator in there).

The flame has nonlinear (like a diode) characteristics that allow it to be very reliably distinguished between a proper gas flame and an open, a short or a resistance due to soot or moisture between the electrode and ground. This is vitally important, because if the gas valve is left on with no flame, the gas will eventually find an ignition source, and Ka BOOM!

Don't fool with this stuff lightly, there is significant redundancy and other considerations required to make a safe, approved and legally acceptable ignition controller.