Fortunately, the mystery really wasn't too hard to crack either I must say. I found a little article from Horticulture and Home Pest News that quickly helped me figure out what was going on. From what I've read there and a few other sites most potato blossoms dry up and fall off without forming these inedible fruits. However, sometimes the blossoms don't just shrivel and instead set fruit. According to the article Yukon Golds set fruit more readily than other varieties and that was exactly the variety of potato involved at the community garden. Mystery solved. Apparently we can just cut them off.
I love my outdoor classroom! I learn something new in the garden nearly every day. Now to just wait until Matt gets home to tell him what I learned today!