The Way of Kings (addressed) |
Words of Radiance (addressed)
Frost is the entity to whom Hoid addresses the letter in The Way of Kings and is also the writer of the second letter in response to that of Hoid in Words of Radiance. Frost has never left Yolen and appears to run the group, The Seventeenth Shard, the members of which include Vao, Temoo and Thinker.
It has been confirmed by Brandon, through the husband of a reader (#46) on tor.com, that the recipient of the original letter is a dragon. Frost is only somewhat older than Hoid. According to Brandon, Frost's longevity is due to the characteristics of his race. 
|“|| "Anyway, there was a God. Adonalsium. I don’t know if it was a force or a being, though I suspect the latter. Sixteen people, together, killed Adonalsium, ripping it apart and dividing its essence between them, becoming the first who Ascended."
" ... Some wished for the power; others saw killing Adonalsium as the only good option left to them. Together they murdered a deity, and became divine themselves."
"So ... my world, and everyone I know, is the creation of a pair of ... half gods?"
"More like fractional gods. And ones with no particular qualifications for deityhood, other than being conniving enough to murder the guy who had the job before." 
The first and the second letters between Hoid and Frost raise questions about the nature of Frost's participation in the conspiracy that shattered Adonalsium. Clearly, Frost had some role in the Shattering:
|“|| "Is not the destruction we have wrought enough?"
"Our interference so far has brought nothing but pain."
"[Rayse] is what we made him to be, old friend."
Brandon has strongly suggested, but never confirmed, that Frost is not a Vessel for a Shard. Unless there's something peculiar about dragons, becoming a Shard's Vessel should have vaporized his body. Yet he's still an "old reptile." But that is probably just a jovial reference.
WoB also says that Frost is "functionally" immortal, "meaning he doesn't age but can be killed." This phrasing suggests that Frost lacks the degree of self-healing capacity that Hoid himself has: Hoid can grow a new head if he were beheaded and is very difficult to kill.
But the first letter states "as you are now essentially immortal, I would guess that wellness on your part is something of a given." This suggests that Frost "now" has sufficient self-healing capacity to be illness-free if not violence free. This new capacity seems to have arisen in connection with the Shattering. Thus ...
1. Because Frost and Hoid are the two oldest Cosmere characters – and presumably had the greatest knowledge of Adonalsium – they knew things the sixteen Vessels didn't know. Perhaps they advised the sixteen just how to Shatter Adonalsium. Maybe they designed the "weapon" that killed It. Neither wanted to become a Vessel himself.
2. Rayse and Bavadin may have persuaded Hoid to help, who then sought out Frost. That would explain why Hoid is so angry at them, "perpetuating [his] grudge" against them. Hoid may feel they betrayed him, especially after Rayse killed Skai and Aona. (And, Tanavast "bought drinks" for Hoid once.)
3. Frost seems to feel guilt over his participation more than anything else. Now he languishes on Yolen, bound by his nonintervention oath.
4. It's possible that the "price" the Vessels paid Frost for his help was his "functional" immortality. Perhaps that's also how Hoid acquired his spectacular Feruchemical ability, as his price. These would have been "paid" after the conspirators ascended.
5. Perhaps Frost's involvement with the Seventeenth Shard is simply as a provider of information. Once Khriss discovered him (and maybe Yolen), he filled in her historical knowledge. It's doubtful that he's an active member, though he appears in ongoing communication with them (or at least Hoid thinks so).