i have no idea what goes on at the farms with 10 million hens as i've never owned or even seen one, the most hens i've seen in a single house were caged hens and that's 180 thousand hens per house, it would take quiet a large farm to have 10 million hens. 
 
each and every farm still uses refrigeration, after the egg is collected you have 3 days (this clock starts the day after the egg is laid) to wash/disinfect and have in refrigeration and 30 days to have them processed and out to the distributor BUT most ( nobody can say for sure what happens at each and every farm though)  of the large caged farms are running and processing continuously and packaged within hours of being laid, they do then go into refrigeration but most are trucked (reefer trailers) out and are on the shelf in local stores within a few days and in some cases it could even be the same day.
 
 
just something to think on, if the farm you mentioned with 10 million hens, on average they are getting over 8 million eggs per day, even with huge warehouse sized cold storage it doesn't take long to run out of space if you're holding that quantity of eggs for any length of time.