Mushroom on the roof

The mushroom in the picture caused a mystery to us. It is attached to the edge of our roof in the backyard. Directly own below are some slippers that has particles like sawdust. We were puzzled where the particles came from. The first thing that came to my mind was the termites that ate the wooden parts of our roof. But the particles are different, it is not powdery and when I touched it, there was a certain similarity with cork.

Those particles that I called sawdust occupied my mind for hours until I spotted that mushroom. Upon closer investigation, there was another mushroom that is now gone. So it might have been pecked by birds and some of the crumbs fell on the slippers down below. That is the only explanation that I can think of and I guess that is enough.