Based on very little data/experience, getting a new general may be part random event and part player action. I got a new general event right after a big victory by one of my general-less armies. The text on the general event led me to believe it was for the guy who lead the general-less army - basically saying "he's got what it takes and we need men like him". I got second new general event that I couldn't associate with any battle action. The text said something about the son of a (disgruntled?) vassal (not a vassal clan) wants to be a general. That sounded more like a random event. It's not much, but I did say it was based on very little data.