I have found one strategy in which 'cutting people off' can be mutually agreeable, maybe this will be helpful to you:
While 'listening' to their predictable words, I try to determine then summarize the vital aspects of it. You already know the 1,000 words they will speak, but can you summarize it well in 50?
This can apply to practical problems that need to be solved, information to be conveyed, or emotions to be heard.
I then politely interrupt ("I just want to be sure I understand, are you saying...") and summarize my predictions.
Many people are delighted to have their own rambling thoughts rendered in a concise, well ordered manner, and/or relieved to have solid confirmation that communication was successful.
Even people who are processing their emotions, and who value 'feeling heard' above all else, will often take pleasure in this response, provided I've done a decent job of capturing the essence of their experience. We can then move on to more nuanced aspects of their emotional difficulties.
Edited to be less long winded. :)
Vice versa, for the past couple of years I've started practicing verbally walking people through my entire thought process from beginning to end. I found that this has been useful in eliminating misunderstandings. It's tiring though, I'll give you that, so I only do it when I believe that it will have long-term positive impact.