I'm glad you mentioned the Mongols, because nearly exactly what you mention exists for them! It's called The Secret History of the Mongols.

It's one book, so it does not constitute an extensive body of texts, but it is regarded as a classic, near-contemporaneous history. The book was commissioned by the Mongol royal family shortly after Genghis Khan's death in 1227, when the Mongol Empire was still very much a going concern. It covers the circumstances of his life, education, unification of the Mongol tribes, conquests, and death.

It was very soon transcribed into Chinese texts and came down into modern times mostly in Chinese sources, with the original Mongolian supplemented by Chinese commentaries. The first English translation was not completed until the mid-twentieth century. Along with a few other later Mongolian works, it's considered a truly classic document that provides a wealth of historical information on the time period, Mongol customs, the life of Genghis Khan, and many other topics. (Personal note: it's also just a great story). Now, given its provenance, it cannot be regarded as necessarily accurate in its entirety. Like any historical document, it must be examined in context and with an understanding of why it was written.

Sources: de Rachewiltz, Igor. "The Secret History of the Mongols." The Mongolia Society Bulletin 9.1 (16 (1970): 55-69.

Atwood, Christopher P. "The Date of the 'Secret History of the Mongols' Reconsidered." Journal of Song-Yuan Studies 37 (2007): 1-48.