How to Correct for a Less-Than-Complete PhD career when wanting to go into academia?
That's great!  That's just how you should feel (1st 2 paragraphs, at least if you're a normal mortal).  It means (i) you're not an arrogant fill-in-your-favorite-anatomical-term-here, and (ii) you're motivated to understand things rather than just get out papers.  This is what a scholar should be.

I felt like there were too many huge gaps in my learning when I was a grad student.  I felt like that when I was a postdoc too.  I felt like that when I was a new tenure track.  I have tenure now, and I still feel that way.  I just learned a lot of basic things over the past year that I can't believe I didn't know 5 years ago.  Part of this is the impostor syndrome, but part of it is just realizing that there there is so much out there to learn.  

There is so much out there, and fields are so rich after centuries/millenia of development that you can only learn bits at a time, and you need to find a balance between solving new problems/getting out papers and learning what has already been discovered.  This is true for bigshots as well as littleshots.  Ravi Vakil (an algebraic geometer at Stanford) says the following, quoted from this MSE answer:


  ...mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". 


I was surprised to hear one of my friends, who is a hotshot and always seemed to know everything, once tell me something like: if we weren't expected to do research, we would spend all of our time just learning the beautiful mathematics that was already discovered. 

At least in math, any postdoc mentor will expect you to have gaps (otherwise you wouldn't need a postdoc), and most mentors will want to you learn a lot of new things, though it is true that some mentors focus on being productive while learning as little new as possible.  So, instead I think you should focus on finding a mentor who you think you can learn a lot from.  Here are 2 things to look for:


They work on things at least somewhat different from, or at least from a different perspective than, what you/your advisor do.  Then you will be forced to learn more new things working with them.
The professor themself seems to know a lot, so you'll learn a lot just from regular interactions with the professor and their colleagues.  This also indicates they're less likely to be of the philosophy of publishing without learning.