Yes we could have a Hungarian Empire but you need a POD as early as 1240. Lets say they can stop the mongol invasion of 1244 before the mongols devastate the hungarian hartlends. Half of their population dosent die and they can remain a grait power of europe. They certenly could stop the otoman advance because they would have 4-5 times of their OTL numbers by that time. So going by that they posibly could have a coast line in the Adreatic and Mediteranian sea. From there they have the chance to conquere Lybia just tu f@@k with the Ottoman. There you go you got now an overseas empire.
That would cause massive, moon-sized ASBs! The Arpad dynasty probably wouldn't die out when it did, the Babenberg's wouldn't either, preventing the Habsburg rise, Habsburg Austria, Habsburg Bohemia and Hungary, and lots of fighting over Austria. Hell, that may end up centralizing the HRE later on and prevent/reverse political fragmentation after the interregnum.
Basically Hungary remains a massive power in the Balkans and probably ends up the defender of Europe against the Ottomans. The Ottomans would be an icebreaker for Hungary, knocking out all the other kingdoms in the area and ending the Byzantine Empire, allowing for Hungary to liberate it all and create an Empire in the region, liberating and holding Constantinople and potentially reuniting parts Christianity. Even if not Hungary just by owning Constantinople and freeing Europe from the scourge of the Ottomans, preventing Austria from becoming the front line repeatedly, would have major gravitas in Europe and potentially end up being an analogue for the Habsburg Empire, but with no fragmented political system and the ability to force Magyar as a language on the Balkan peoples for hundreds of years, while also controlling major amounts of food producing areas and oil reserves. I wouldn't be surprised if Hungary even got to help carve up Poland later on, getting Galicia and their oil/farm land.
Of course this assumes that all goes well after the Mongols spare Hungary, which is not certain, but is possible.