Looks like the bin of Horse Manure got a bit cooked!
We have white stuff in our compost which looks to me like ashes. It shows up when horse manure or lots of grass clippings are used. I would say when there is high nitrogen content. Usually I've seen it after the pile got really hot (Can't hold onto the rebar that was embedded in the compost). Is it Mycorrhizae or maybe ashes?
What the Ashes Probably Are?
For sure the ashes are not mycorrhizae. Mycorrhizal fungi don't live in compost piles. Compost piles are hosts to all kinds of decomposers but that type of fungi only lives embedded in living roots.
I think this is probably actinobacteria aka actinomycetes. They are the organism that makes soil smell so good. And they seem to be able to tolerate high temperatures.
I would suggest that when you have temperatures so high you can't hold the rebar that that is too high. Turn the pile and add water to cool things down and you'll get even better compost with more biodiversity in it.
Thanks for your question. I do have a page on actinomycetees here on the site you might like. Check it out. http://www.the-compost-gardener.com/actinomycetes.html