Fireproof Lizard Roundup
At the Southern Mine, an NPC named Kima is looking for Fireproof Lizards. Talk to him and he will ask that you bring him ten of those sneaky little critters.
Fireproof Lizards are all over the Southern Mine. If you are stealthy, you can probably round up ten in just a few minutes. You will have to walk quietly and slowly up to the lizards out in the open or they will scatter and then vanish.
Additionally, lift up every rock you see that gives you a prompt to lift it. Fireproof Lizards are under almost all of them. Here's how you can grab them. Press [A] to lift the rock. Immediately after it's been raised, press [B] so that Link can drop the rock. Then immediately press [A] again, even before Link finishes the animation for dropping the rock. Otherwise, the Fireproof Lizard will run away and be too far for you to grab it.
For an easier time hunting, especially if you leave the Southern Mines but want to continue looking, get a photo of a Fireproof Lizard and use your upgraded Sheikah Sensor to target it. Learn more about how to upgrade your Sheikah Slate here.
Once you have ten, give them to Kima and he will give you the Flamebreaker Armor (specifically, the chest piece). Even if you already have it. For some of you, this will be worth it because you don't have it yet, but for others, it will just be a duplicate piece of armor. Sorry not sorry?
Note: In the future, you can also bring three Fireproof Lizards to Kima and he will give you 20 Rupees. Whether or not that's worth the trade is up to you, considering that Fireproof Lizards are also necessary for armor upgrades.