I think drawing may be a similar skill. I can't actually draw well, but my model of how one does this is by not interpreting your visual field by the laws of perspective and instead making lines that correspond to your actual sensations. This is why sometimes one sees it recommended, as a learning technique, to put a photo upside down and then try drawing that. It forces you to pay attention to the lines instead of performing the visual parsing step one ordinarily does. (The other half of drawing is causing the lines to show up on the page in the place where you want them to; this is not part of the analogy.)
One could consider this Looking at one's raw visual phenomena. The skill described in the OP would be Looking at one's own mental phenomena, a layer or two further up the stack from the raw visual phenomena. It's possibly important not to think one has the generalized skill of looking if one just knows how to draw (it may block you from actually learning how to Look). However, if this analogy holds as strongly as I think it does, then one ought to be able to learn both skills with similar techniques?
For drawing that implies perhaps spending time studying your visual field without parsing it may improve your skill, even without practice (clearly at some point you also have to learn the skill of putting lines where you want them). This sounds a lot like meditation, but applied to different sensory input. This seems like a testable prediciton of this hypothesis/theory.
Then again, I might be normie-splaining; I don't draw well, and I either haven't had the experience described in the OP or have but don't consider it remarkable (honestly can't quite tell). But this seems like a stronger analogy than any I have had heard.
Not all domains have obvious outputs. It's entirely plausible to me that there is no equivalent of "drawing" for the skill of Looking-at-mental-phenomena.