An Electronic Expansion Valve is basically a stepper motor driven valve which regulates the refrigerant gas flow into the cold part of your cooling system. (evaporator)
this will replace the usual capillary tubing and allows to fix an exact cooling temperature.
this stepper motor can be controlled by an arduino and be automatically adjusted to your desired temperature if you link up a thermocouple and a pressure sensor.
did anyone build their own fridge compressor from scratch with automatic controll?
solder up the evaporator (cold) and the condenser (hot) with copper tubing and an air+moisture filter, (optional a compressor silencer)
vacuum down system, flush with nitrogen , repeat 2x, then fill it with matching refrigerant gas.
since a common expansion valve uses a temperature feedback and diaphragm to regulate, it does not seem that adding motors and electronics would add any benefits,.
it would add complexity, expense and uncertainty of longevity (possible premature failure)
I have not seen any posts on this topic before.
I would suggest you search the forum
and if you search the web, add 'arduino' to your search
arduino expansion valve
maybe this is too complicated for an arudino beginner and using a capillary tube at first is better with a simple on/off switch for the compressor which is controlled by 1x temperature probe at the 'cool side' by an arduino.
once x temperature is reached turn off compressor, maybe PID controlled...