Your challenge, with a POD after the collapse of the Ilkhanate in 1335, have Anatolia come to be dominated by a dynasty that does roughly the same things as the Ottomans (ie, capturing Constantinople, menacing Vienna)...but is Shia (and yes, I realize a Shia dynasty will obviously have a different name!), with present day *Turkey being a majority Shia country. Meanwhile, have Iran stay Sunni.