Here's another goodie from the alternate-history forum I've been a member of since high school--"TLIAW: Kayser-I Rum." For the record, TLIAW means "timeline in a week." Basically bang out a complete alternate timeline in a short period rather than just theoretical discussion, which most threads in the main forum consist of.
The point of divergence in this scenario is that the Mongol warlord Tamerlane, rather than invading Asia Minor (where in our history he defeated the Ottoman sultan and kicked off a four-sided Ottoman civil war), instead wages war against the Mameluke Sultanate in Egypt after the latter launches a preemptive strike on his forces in Mesopotmaia. Consequently, Constantinople falls to the Ottomans under Bayezid I "The Thunderbolt" in 1403. This in turn provokes a new crusade by various European Christian states, a crusade that ends badly for them. After Bayezid's death, his son Suleiman takes the Ottoman throne and openly wonders, "If we have the Second Rome, why not the first?"
Highlights from this timeline include:
*The Ottomans forcing the Venetians into submission after a protracted naval war. This has its benefits, including the growing Ottoman Empire becoming a massive Venetian free-trade zone.
*The Ottoman Empire remains focused on expanding into Europe and stays out of the greater Middle East for much longer than it did in our history. Suleiman follows Roman precedents more strongly--he's basically a Muslim Roman emperor and is compared to Marcus Aurelius as well as the Caliph Umar. I imagine this timeline's Ottoman Empire is going to be much more "European" in nature even if it is a Muslim state.
*Extensive Ottoman expansion on the Italian mainland. Per my earlier point, Suleiman even recruits Eastern Orthodox soldiers for the war by selling it as a crusade in retaliation for past Catholic misdeeds. The Duchy of Milan takes advantage of this to expand its land empire.
*France becomes a centralized state with a standing professional army much, much earlier than in our history. Although many historians attribute French national identity to the French Revolution, my reading of Barbara Tuchman's A Distant Mirror: The Calamitous 14th Century would suggest an earlier origin (at least for the upper/educated classes), the 100 Years War. This time around, it's cranked up.
*There's a war between the Ottoman Empire and Switzerland of all places. It goes rather differently than one might think.
The timeline hasn't been updated in awhile, so it's probably dead, but it's a pretty interesting read regardless. Enjoy!