The case in question is episode 10 of Fullmetal Alchemist
Fullmetal Alchemist actually features quite a lot of mysteries and investigations like fraud and serial killers, not to mention the overarching search for the Philosopher’s Stone. Still, I wouldn’t go so far as to call it a detective series.
But one favorite trope appears in this early episode, the calling card dropping phantom thief, in this case she is called Psiren. Ed and Al happen upon in one of the town’s they are passing through. It is revealed quickly in a fight with Ed that Psiren is using alchemy for her capers.
There aren’t a lot of suspects in the episode so it is not too hard to deduce that Clara the nurse is indeed Psiren and it doesn’t take Ed long to figure it out either. Most of the episode is about chasing her and figuring out why she is stealing.
Clara spins quite a story to the boys about wanting to help the hospital which is losing its funding. However, it actually does close much to the shock of Ed and Al! Then they find Psiren all about town at various other in-need establishments to cover her motives. I rather liked that in the end Psiren escapes to steal another day.