BLUF: If a direct does not engage in the conversation, how hard (and many weeks) do I keep bearing the load of working the relationship? I have been told she is somewhat better with team members she's known awhile. This extreme introversion has also affected her career - though I don't think anyone has ever told her that directly. Do I give her feedback on the O3 or that she needs to overcome this to advance?
Details: I've been managing a group for about 6 weeks. One report has had 9 managers in 7 years. The person is an extremely high "C". We have had two initial 1 hour meetings, 1 face-to-face (the report is remote from me) and 4 O3's.
Every discussion (if you can call it that) is a struggle on my part. I try to understand that the report is a high "C" and not comfortable with chatting. But I do need the report to take some ownership of her part of the meeting. By specifically asking, I have dragged a few things out of her. But I come out of each meeting feeling exhausted with the effort of a completely one sided conversation.
I'm thinking of giving the following feedback if I get one word answers to every question I ask. 'Can I give you some feedback? When you're input to our one-on-one's are that "I'm on schedule. I'm fine. I completed my x, y and z.", one word answers to my questions and you're part of the O3 is always less than 2 minutes, it is hard for me to judge where you are. What can you do about it?'
I've thought about giving her topics that she has to discuss with me like telling me more about her area of work. But again, that seems like me doing all the work. I realize that she is extremely introverted, but that does affect her ability for growth on the team.
My other thought is that I was told by my boss that the direct asked why she hasn't been promoted. My boss put in one for her during the gap before I was her manager. That should go through soon. Her ability to advance is definitely limited by being so introverted. Technical skill takes her to a level but the next level requires working across teams and having a larger impact. I could talk to her about coaching on that to move up in her career.
Which do you think is the better alternative? Or something totally different?

O3 when report doesn't engage
You can't make 'em talk if they don't want to. Trying to force it won't get anyone anywhere. Like you said, she's a C.
If you simply tell her she has to be more outgoing you're probably not going to get anywhere. After all, what does that have to do with getting work done, right? :)
I think you're on track with the coaching idea. Coach her on career advancement, not interpersonal skills. Have her do some research on skills she'll need to succeed in management. If she doesn't get there on her own add in relationships with peers and directs. Then coach coach coach. Give her feedback back into her success with the coaching.
Basically, she has to see why she should bother with all this people stuff before she's going to bother with it.
O3 when report doesn't engage
Ask questions about work that require her to describe what she's doing in more detail. Try something like "how did you overcome this design challenge?" or "What did the customer say to express their unhappiness and what did you say in return?" (Don't know what your team does....)
If your questions are vague, the answers can be terse. Yes, you will take the lead in the conversation. Have you listened to the "Everyday conversations" podcast of Oct 1, 2007?
Don't press on personal life. Stick to work issues. When it's your turn, you can talk about your stuff (including personal). I have one direct who never discusses personal life, but the O3's are good.
John
O3 when report doesn't engage
It seems to me that 6 weeks is not that long. That's just six 30-minute meetings for your introverted direct to get comfortable with you, assuming you stared the first week. One of my directs and skips took nearly a year to get really comfortable. That's not to say it [i]should[/i] take that long, just offering a point of comparison.
Because she's used to having a new manager every year (or more!), she probably finds it very difficult to trust that you'll be around for awhile. The stress of new bosses with new demands can get to anyone. For a high C who needs precision and certainty before acting, it must be agonizing.
Another thing to keep in mind -- introverts experience more physiological arousal than extroverts. That is, they experience more physical symptoms of anxiety compared to extroverts in the same situations. A meeting with a new boss can be quite stressful for someone whose baseline arousal is already high. If she's sitting there with a racing heart or perspiration trickling down her back, it may be difficult for her to "loosen up" and be more conversational with you. Not saying she won't get there, but it might take longer than you'd like if it's going against her biological make-up.
Have you given this direct lots and lots of positive feedback already? It seems this would begin to make her more comfortable with you as she learns to trust that you notice and value what she does well.
I'm guessing most of the positive feedback she has received in the past was based on her high-C behaviours (accuracy, thoroughness, etc). As she gets more comfortable receiving positive feedback from you, start to add in positive feedback about aspects of her social interaction with you or others. The positives will probably be really small at first, which is okay. Show that you value her efforts in that domain before you start pushing for improvements that are going to be challenging for her.