There are no rules in Rise of Tiamat (or any other source) for how to put the mask together or how to take it apart. The description only indicates what happens if you do. Silly, perhaps, but that's what's written.
In an non-Adventurers League game, your GM can allow or disallow breaking up the mask as he likes.
Adventurers League Considerations
For Adventurers League characters prior to Season 8, expect table variation. While WoTC has released stats for the individual masks, only the Black Dragon Mask shows up in a "Treasure" entry, which means it's the only lootable one in RoT. For AL purposes a table DM would be entirely correct in a belief that the individual masks do not exist and are not lootable.
When Season 8 starts at the end of August 2018, the Mask of the Dragon Queen is on the problematic items list. If the character has finished Rise of Tiamat with a mask, it is converted to Treasure Points. If they're still in the adventure, it becomes a story item that will be gone at the end of the adventure.