Moscow to Beijing / train through Mongolia.

Haven't done it ourselves, but had some friends who did the trip with Road Scholar a few years ago: https://www.roadscholar.org/find-an-adventure/19580/trans-siberian-railway-from-beijing-to-moscow. They raved about both the company and the experience.
The detailed itinerary presented here might at least provide a starting point for your research if you opt to do this independently.

Before you try this trip, I suggest reading this book:
https://www.amazon.com/Nothing-True-Everything-Possible-Surreal/dp/1610394550

The last part of the book describes a trip on the Trans-Siberian railway.

Or, for a more down to earth take on it, check out Lisa Dickey's Bears in the Streets (www.amazon.com/Bears-Streets-Journeys-across-Changing/dp/1250092299). It only deals with Vladivostok to SpB leg of the line, but is pretty good at building a perception of TransSib as a mode of transportation between destinations, rather than a destination per se.
Unfortunately, in the vast majority of cases "I did TranSiberian" is akin to "I survived a Trans-Atlantic flight".