John Workman posted a comment about the mystery object that might go a long way toward a solution.

He advised checking out the following Website:

http://botit.botany.wisc.edu/toms_fungi/calvatia2.html

After a little exploring, I found a Spanish website that has a variation that looks a lot like our guy. Link is here. (Not a lot of info, but very similar looking puffball.)

Thank you, John, and all the (somewhat vaguer) fungi proponents… !