Maybe we can have a Mongol conquest of Europe. Following the same tactic elsewhere, the Khan decides to co-ope an indigenous institution to grant the conquest more legitimacy. Since Europe, unlike China or Iran, had no tradition of centralized Empire, the Khan decides to use the one pan-European institution - the church.
A relative is sent to Rome, ostensibly as a legation to the Papal States. The legation lasts several months where the Pope is advised to abdicate on the grounds of ill health, while the relative somehow undergoes a miraculous Paul-like conversion to the one and true Catholic faith.
The next Papal Conclave, held while Mongol cavalry are parked outside Rome's city walls, elects the Mongol as Pope. The new Pope then floats a theory about how he's Prester John and pays lip service to liberating the Holy Land from the infidels, but quietly discards it later.
The new Mongol Pope then swiftly packs the church hierarchy throughout Europe with his friends and allies. Of course, these are all supported by his fearsome cavalry, who are now flying banners with crosses. All of Europe's monarchs are at least nominally vassals of the Holy Roman Empire, whose Emperor is crowned by the Pope.
Long term legacy: even after the Mongols collapse, the Catholic Church becomes even more cesaropapist than its eastern brother.