Garbage Disposal - Water Shoots Up Non-Disposal Side
Replace that 4 way junction with two wyes offset from each other (you might drop the disposal trap a few inches to achieve this) and extend the aav arm as high as possible.
The pressure from the disposal may be forcing the aav shut, and between the poor configuration of the hub and the resultant lack of venting the opposite sink is the path of least resistance.
Yes you will have to chop that monster out.  If you can't configure two offset wyes at least get a double wye hub and raise the aav.