Guys, guys:
The entire story (TSRoST and the main trilogy), Auri demonstrates a knack for naming things. She names all of the rooms in the Underthing with names that fit them perfectly. She names Foxen and Fulcrum. She refuses to call something by a name until she feels she has discerned what it wants to be called, what it should be called. She's obsessive about it, and she's really good at it. "Hey, she might even have some Naming aptitude herself!" you might think. Buddy, you don't know the half of it.
The first part of your answer comes when she wants to make Kvothe a candle, but she knows that rendering the laurel fruit she wants to use would take hours that she doesn't have. Even if she rushed herself, she wouldn't have time.
It was just as Mandrag said: Nine tenths of alchemy was chemistry, and nine tenths of chemistry was waiting.
The other piece? That slender tenth part of a tenth? The heart of alchemy was something Auri had learned long ago. She'd studied it before she came to understand the true shape of the world. Before she knew the key to being small.
Oh yes. She'd learned her craft. She knew its hidden roads and secrets. All the subtle, sweet, and coaxing ways that made one skilled within the art. So many different ways. Some folk inscribed, described. There were symbols, signifiers. Byne and binding. Formulae. Machineries of maths...
But now she knew so much more than that. So much of what she'd thought was truth before was merely tricks. No more than clever ways of speaking to the world. They were bargaining. A plea. A call. A cry.
But underneath, there was a secret deep within the hidden heart of things. Mandrag never told her that. She did not think he knew. Auri found that secret for herself.
She knew the true shape of the world. All else was shadow and the sound of distant drums.
Auri nodded to herself. Her tiny face was grave. She scooped the waxy fine-ground fruit into a sieve and set the sieve atop a gather jar.
She closed her eyes. She drew her shoulders back. She took a slow and steady breath.
There was a tension in the air. A weight. A wait. There was no wind. She did not speak. The world grew stretched and tight.
Auri drew a breath and opened up her eyes.
Auri was urchin small. Her tiny feet upon the stone were bare.
Auri stood, and in the circle of her golden hair she grinned and brought the weight of her desire down full upon the world.
And all things shook. And all things knew her will. And all things bent to please her.
It was not long before Auri returned to Mantle with a sorrel colored candle pressed with lavender. It was a perfect thing.
(pp. 143-145)
So, what exactly happened in the above paragraph is something I've thought about a lot since I read it. Whether it means she's a Shaper, has somehow discovered Grammarie on her own despite not being Fae, is actually Fae, or what, I'm still confused about. What I do know is that she is a lot more powerful than I suspected, and more than Kvothe or anyone else suspects too.
Here's the second part of your answer. It's the last lines of the book. You're not the first person I've heard ask this, and the rest of this thread is indication that you're far from the only one with this question, so I'm not saying you're a careless reader or anything, but still, I'm like "How in the living fuck did anyone not read this and have his or her mouth knocked agape by the force of its fucking import?"
First was his clever candle, all Taborlin. All warm and stuffed with poetry and dreams.
Second was a proper place. A shelf where he could put his heart. A bed to sleep. Nothing could harm him here.
And the third thing? Well... She ducked her face and felt a slow flush climb her cheeks...
[...] [Aside: I think Rothfuss is using that line to goad us into thinking it's sex. It's not sex. I mean, it might involve sex. But not primarily. Keep reading.]
Still, there was the third thing. This time Auri did not blush. She smiled.
[...]
Excitement bubbling up inside of her, Auri looked at his bed. His blanket. His bedshelf with the tiny Amyr waiting there to guard him.
It was perfect. It was right. It was a start. He would need a place someday, and it was here all ready for him. Someday he would come, and she would tend to him. Someday he would be the one all eggshell hollow empty in the dark.
And then... Auri smiled. Not for herself. No. Not ever for herself. She must stay small and tucked away, well-hidden from the world.
But for him it was a different thing entire. For him she would bring forth all her desire. She would call up all her cunning and her craft. Then she would make a name for him.
Auri spun about three times. She smelled the air. She grinned. All around her everything was proper true. She knew exactly where she was. She was exactly where she ought to be.
(pp. 145-147)
After whatever happens to Kvothe that makes him feel dead inside - losing Denna or whatever it is - he comes to Auri all eggshell hollow empty in the dark, seeking asylum. She gives him a safe place to stay. And she gives him a new name.
She's the reason he's the way he is when Chronicler finds him.