I had assumed it was the work of some modern artist as it had a "made in Italy" stamp on the back with an artist's name, "Milanaccio". Some googling though revealed that the artist in question specialized in making replicas of museum pieces. I called the Morgan Library and they confirmed that my tile was in fact a replica of a piece in their collection, but since they no longer sell this item they couldn't say for certain what exactly it was a replica of.
So I looked around on their site and found some pdf advertisements of past exhibits. With some keyword searches and help from google, I found an advertisement with a picture of the original item and a collection number. Using this, I tracked down the origins of the piece.
My tile is actually an imprint of a seal that would have been rolled in clay to leave an impression. The seal was found in Persia, and imprints found on a certain tablet dates the seal to the time of Darius and Xerxes (Achaemenid period, ca. 550–330 BC). If I'm not mistaken, this would place it around the time of Esther in the biblical narrative. How cool is that?