As much as the idea is fascinating (and it really is) the distance is incredible. This would truly be "World Empire" status power here.
First you'd need to address the security of the Hexi Corridor, at the very least. This might mean some sort of persistent relationship between China and the Steppe, be it like the Xiongnu, or something altogether different, further, the same between China and Tibet. Otherwise that corridor snaps everything in two rapidly, which could be a disaster.
However, if we set a precondition of a Tibetan conquest, and negotiated relationships with the Steppe, we could have something here. Since a lot of the personal loyalties of new steppe leaders relied on personal success, China could effectively supply and endorse them riding westward WITHIN China, to then raid Central Asia. That is one difficult treaty to maintain, with details to resolve, but lets also patch that issue.
We now have a China with relatively secure logistics, but not always as the scale required, which means the Tarim Basin needs to be able to support any army, or the Chinese will be establishing huge food stores that then need to be guarded. Which is getting expensive.
Further, we then take Central Asia - it's demographic core in the Fergana valley being the focus. This means we now have two regions highly exposed to the steppe, which still needs some sort of resolution.
Then the Abbasids. Perhaps your resolution is to effectively ceremonialise the inheritance of power with the steppe groups to the extent that their leadership challenges happen on the edge of Central Asia and the Abbasids, in which case you've got the far west of the Chinese Empire effectively 100% at war with the Abbasids at all time. Ignoring that this then means perpetual war with a significant trade partner, even if indirect trade, a counter-attack is easier for the Abbasids than the Chinese.
So you need to figure out a way to secure Persia. Now this is easier, as once you've conquered Persia, you could settle it en-masse. The Chinese certainly had the demographic advantages there, but your logistics are still perilous by land or sea (because well, India and Indonesia can dip their fingers in this pie.)
However, if the Chinese can somehow maintain their Western Protectorate in Persia, then that can be the base of operations for a truly collossal endeavour.
The final point of this is that to perform the challenge you want, you have to address multiple sets of major geopolitical challenges. Not impossible, but you're basically looking at a China-built Steppe coalition, conquest of Tibet, Conquest of Persia, Conquest of Mesopotamia, THEN Conquest of the Romans. This is Genghis Khan/Caliphate level stuff here. Not to mention the need to never suffer anything like An Lushan.
I don't really see it as plausible, not unless the Tang Dynasty is radically more successful than it already was. If you want the Tang invading Rome, you might need to have them become truly the World Empire, by conquering Indonesia, and potentially even India. At which point the sheer economic power in play, and the resources at hand make the task much easier. Especially since this secures naval logistics. But even then, you have the internal issues to resolve.