Easily the Umayyad Caliphate. No other empire has even come close in terms of size, long-term impact on the areas ruled, and total unrivalled dominance. They even defeated the Chinese in battle, while at the other end of the world their armies were active in northern France. This is simply incredible and even more impressive considering the technology of the time.
The Umayyad were defeated by the Tang dynasty at several engagements and by the Karluk allies of the Tang. Umayyad incursions in the region were effective, but the Umayyad had yet to vault if you will the greatest test in the region, the Tang and Tibetans. In Bactria, the Umayyad had outmanned and defeated the Tibetans in Kabul and subjugated the Shahi states, but the Umayyad had failed to consolodate the region. In Zabul, the Umayyad were defeated twice by the Zabul people in the hill country of modern southern Afghanistan and once, the invasion the Umayyad sent (ordered by Hajjaj ibn Yusf, the governor of Iraq and during the period of Caliph al-Walid) to capture Zabul, the famous 'Peacock Army', came under Khawarij leadership and rebelled and would attack the Umayyad. Furthermore, the area of Ferghana-Sogdia remained mildly outside Umayyad domains, while Kwarezm had more or less been subjugated by the Umayyad. In short, the Umayyad had not fully acquired the hegemony in the east, that the Abbasids would gain after defeating the Tang at Talas and thereafter, subjugating its internal foes in the east along with the conquest of Zabul and the incursions into Hindustan.
The Umayyad had also not successfully dealt with the Khazars to an adequate level, the Khazars had dealt deadly wounds to the Umayyad while the Umayyad could in turn inflict superficial blows as the Khazar avoided direct engagement in the Umayyad invasion of Khazaria in retaliation for the Khazar invasion of Iraq. So, the Umayyad was surely a great power for its period, but to say that it was greater than say the Roman Empire in terms of comparisons to its nearby counterparts, is to me lacking. Though, you are right in the statement that the Umayyad were a powerful force, the Umayyad especially should be remembered for their effective usage of decentralized frontier wars that allowed for the command structure of the Islamic world to expand in such far distances and produced the phenomena that you mention of warring against an expansionist Tang while also battling the Franks in Gaul. The reason is, the two armies at separate ends of dar al-Islam (abode of Islam) were not really in connection in the sense of modern armies, but were individual actors of the same political leader, who gave them free reign to do as they pleased. The Umayyad caliphal period itself was most focused on its war at sea and the war with Byzantium, it was unable to defeat Byzantium fully due mostly to luck, poor military recruitment choices by the Umayyad (they used conscripts, most of whom refused to fight, the standard in Islamic history is to use only those who are truly willing to fight, not use those who have questionable statuses) and Greek fire as it is called. However, Umayyad adventurism at sea was truly novel and much of the Umayyad success in this period against Byzantine and Latin interests can be owed to its skill at sea, which was unthinkable to the earlier Arabs. It is interesting how this naval forays lessen with the Abbasid period, no doubt this is due to the Abbasids by their outlook and positioning, having an eastern and inland perception of geopolitics from Baghdad, while the Syrian Umayyads had a taste for the Mediterranean.