Well, a google on Milk+fig tree turned up a variety of things, but it seems the milk is what the fig tree has instead of sap. So, one would conclude that you'd milk it when the sap rises. But then again, could be fall. My ficus (benjamina I think - the one everyone has in the house) gives milk when ever you injure it. It's pretty prolific at all times, so I would imagine a fig outside would "give" when the weather is warmer. I did see several things about the milk removing warts and being used to render cheese (?). Heck, start hacking and see for yourself - haha - poor tree! There probably IS an art to it though. Seems that the easier it seems, the more intricate it is. Since plants do different things at different times, it would only follow that the milk at certain times would have certain properties.
My aunt had a great fig tree on her property. Beautiful fruit. I didn't appreciate it then of course. It just made me have to pick figs for fig preserves that I didn't appreciate either! Oddly, I think that I spotted one that has taken root in my garden. I am going to investigate it to make sure that's what it is. It would be really neat if it is. Perhaps Auntie has left me a gift!