A Greater Dar al-Islam and an Ever Shrinking Dar al-HarbThis timeline is partly inspired by the Muslim World: The True Faith timeline and by an in-development mod for Hearts of Iron 4 called Bayt al-Hikmah, though with my own views and interpretations on how an Islam-dominated world would look. The point of divergence is the death of Charles Martel at the siege of Avignon in 737; from my reading, this is a perfect point to break the Carolingian empire before it began and cause a lot infighting in Francia.
The point of this timeline is not necessarily to create a Muslim Europe (though that may end up happening) but instead to prevent Western Europe from becoming the world's dominant region, and so keep dominance in the Mediterranean and the Middle East.
Feel free to comment and offer criticism on the timeline.
A note on naming: I will try to maintain a consistent approach to names of people and places but this of course can be tricky when dealing with transliteration from other scripts.