I applied to a then nascent part of McKinsey (that is now very mature) called Digital McKinsey, as someone who could help clients build and transform engineering and product organisations. To be a CTO/VP Eng for hire, almost. Then I slowly moved more and more into the non-tech topics.
I was there for almost exactly four years. Learned all kinds of things, but most importantly how to communicate and structure my thinking in a way that works with people senior people from non-tech backgrounds.
* Empathy with the folks who's thinking was alien to me previously. I learned through doing their job. I could communicate with them and work in their context in a way that allows me much more leverage and ability to have impact with engineering.
* I learned the semantics and pragmatics of running a large organisation which gave me the lexicon and confidence to be taken seriously doing it (therefore more leverage) and taught me lots of small but important tactical lessons (e.g. how to manage difficult shareholders)
* How to communicate complex technical subjects to people who either don't/can't/won't understand them.
* A HUGE network
I went back because I missed it. I wanted to do consulting as a way of getting better at being an engineering leader, not leaving it forever. I was offered a few things at VC/PE firms, or in typical 'business' roles, but none of them really appealed. Not my passion.
I had never thought of the big 3 as possible route post-sale. Sale is not on the radar atm so back to my daily operational duties in my small yet thriving company :-)
I realised that (as CTO) the main area I couldn't contribute to was 'the business' of running a successful company. So I decided to remedy that, I joined McKinsey and left last year as an Associate Partner. I'm now the VP Engineering at a fast growing tech company (250+ people) and the impact I can have with technology is an order of magnitude greater than before because of the intervening experience