I started running 'Talk with the Experts' sessions at SitePoint about 6 years ago, and spent years refining the process.
The platform was the trickiest bit. The key factors were low barrier to (or guest) entry (people didn't want to have to sign up), text based (the majority of experts didn't want to appear on video), and ease of exporting a transcript. I trialled several text-based chat apps, none of which were very good. Some threw errors all the time, some didn't allow you to @tag people, others didn't allow exporting of transcripts. I ended up designing a custom platform which we built based on Kiwi IRC. It was brilliant, but has now been retired as they don't have the resource to run them.
Even with a great platform, there were issues of time zone (with an international readership it can be tricky to find a time that suits your greater audience), reliability of experts etc.
At UXMastery I currently run a series called 'Ask the UXperts'. I've gone back to basics, using a Campfire chatroom. It works fine, but is very basic and isn't perpetual, meaning that I have to continuously jump in with 'Welcome to those of you that have just joined us...' (which I then have to edit out of the transcript). I always run a Skype text chat session in tandem with the expert, so that I can queue questions for them. That way they can concentrate on answering, rather than having to filter through the noise. You'll find plenty of transcripts of sessions on uxmastery.com if you're interested.
Main lessons learned:
1. Don't run them to a schedule. If you commit to doing 1 per week/fortnight/month you put undue pressure on yourself and end up settling for less than ideal candidates.
2. Get some seed questions off your expert before you start so that if things are quiet (or going off topic) you can steer it in the right direction.
3. Have someone on backup if things get busy. Our sessions ranged between 25-250 members and I never knew ahead of time how busy it would be.
4. Advertise the session well. We put a banner up at UXM (there is currently a live one up at community.uxmastery.com), as well as the expected social media stuff.
In the past I have attempted putting up a forum thread first to collect questions, or to give people that won't be able to make it the opportunity to ask them, but that has never been especially successful.