Liquids expand when heated, that's why it gets fuller.
Have a look at this PAIR valve removal thread, you will need block off plates if you want to remove it permanently, but they aren't expensive.
Once you have the valve out (top of pic) you will see three pipes:
To test it unplug and remove it from the bike then measure the resistance of the solenoid coil, it should be between 20 and 24 ohms.
Next try blowing air from the long pipe through to the shorter ones, no air should come out.
Now apply power (12V from the battery is just fine) to the solenoid (polarity doesn't matter) and try blowing again, this time air should be able to flow freely to both of the short pipes.
If all that goes well then the valve is fine and can be reinstalled...
Next thing to check that system out is to leave the solenoid unplugged, but everything else installed correctly. With this done start the bike up and see how she idles, if its all good your ecu is opening the PAIR valve when it shouldn't be. If its still stalling then you will need to plug the PAIR valve back in.
After all that I would be checking the ignition timing, which should be 15 degrees BTDC (F mark).... and the hard part is that this is at idle, however you should be able to see it either advancing or retarding way to much to run if the timing is out.