The definition of doujinshi is self published works and a lot of them happen to be fancomics and based on other series. Doujinshi does not have to be just comics, as there are some that are novels, artbooks, also there's also doujin music which is a whole another world you should get into. When a lot of people think doujinshi, they think of h*ntai manga, but thats not really accurate. All the stuff I have linked here are at most PG-13.
GeneralTeddie Comic A sweet comic about Teddie's loneliness.
Teddie Hanamura Another sweet comic about Teddie, this time about family.
Puella Magi Madoka Magica
Madoka Kaname x Homura AkemiKiss or Chocolate A short and sweet (pun intended) Madohomu comic.
Edward Elric x Winry RockbellYou see, actually I'm... A post-canon doujin about the life of Edward and Winry's married life.
ZinesSoul Binds A zine about Edward and Alphonse's brotherhood.
Royal Pain A zine dedicated to best boy Ling.