'Can you find the ‘thing’ that is doing the choosing?'
There is no thing which is doing the choosing. the choosing happens automatically. There is no need to listen to thoughts because choices are done without thinking. They are basically even before thinking happens.
However, there are many things influencing choice. Like the way the apparent self is brought up, physical abilities, what kind of information one has, what kind of environment you live in, the past of the apparent self, etc. etc. Those things influence the choices. But there is no one doing the choice. No one sitting there and making the choice. The choice happens by itself.
'What is the actual experience of choice?',
The actual experience in choice is that there is no control nor is there free will. Choices just appear and disappear. They come and go, just like thoughts. There is nothing particular about it. There is no way to control thoughts.
'What would have choice?'
Don't understand the question, is there a word missing?
Please explain how the notion separate self was found to be just an illusion--in your own direct experience. What is the first memory of how it got a foothold, and how did it present itself from there, as you experienced it?
The first memory was about 3 1/2 months ago when I was reading a neo-advaita book and the chapter was about thoughts and no-ego. I was studying and practising Buddhism for the 12months before. During my studies of Buddhism I hardly came across the discussion of "no-ego". Thus, I abandoned Buddhism and was seeking for other material and instantly came across neo-advaita. This body-mind of Georg was highly ready for neo-advaita talk and when I read about it I instantly felt relieved from many burdens.
The first memory was about reading the chapter and then visiting my parents watching television. I couldn't help myself and laughed for about 2 hours straight. I was laughing about the belief that thoughts are us. That we identify ourselves by thoughts. That there is something as an identity caused by thoughts. This was extremely hilarious and as I mentioned above, I was laughing for two hours!!
Well, after that night I fell back into old beliefs and started identifying myself again by the thoughts happening in this body-mind. Neo-Advaita didn't help me onwards. Thus, I found Joseph and James Swartz who helped me to look through this illusion.