Legendary battles are tiles that have a face on them. Unlike most tiles, legendary battle tiles have multiple stages per tile. In each one you have to play as a specific officer (the 2nd player if you are playing coop can be whoever).
Since there are 62 characters, and at least 2 stages per character, there are WAY more than 10 legendary battle stages to finish, so this will be easy.
You have to finish the legendary battles of most characters in order to use them in conquest mode (some unlock by playing the story modes, but many of them have to be unlocked this way), so you'll probably get this without even trying.