Inspired by this remark of KMP, below
The only one which is really close and that is in 1808. There are two male members of the dynasty: Mahmud II (Sultan) and Mustafa IV (deposed but alive). In 1808, the Janissaries rioted in town after indirectly killing Grand Vizier Alemdar Mustafa Pasha, trying to restore Mustafa IV on the throne. During the riot, the city was in chaos. The pro-Mustafa Janissaries fought against the pro-Mahmud forces and the navy.
Now... in this confusion Mahmud II ordered the execution of his brother Mustafa. If the Janissaries break in the palace and kill Mahmud while the executioners kill Mustafa without both having knowledge of each other... this would mean the dynasty is extinct. And the remaining Giray family can be placed on the throne but the semi indepedent pasha's will most likely refuse to acknowledge. And may break apart...
This is to me the only best* possibility
I was going to call it an Ottoman "circular firing squad", but "everybody must get choked" seems more appropriate since strangulation was the preferred method of execution.
Now it's not the first time this scenario has been considered, but I recently learned that the Ottoman Empire was in a state of war with both Britain and Russia at this time.
Will prospective Ottoman weakness motivate the British to try to attack Egypt or any other Ottoman territory again? They had already tried and failed in expeditions against the straits and then Alexandria in 1807.
What will the French military mission of Horace Sebastiani in Constantinople try to do, and who will it line up with among non-Ottoman power contenders. The Sebastiani mission had participated in organizing the defense of the straits from the British earlier.
Will the Russians increase the scope of their campaigns into OE territory beyond what they did in OTL 1808-1812?
How much of a chance could this situation provide for Britain and/or Russia to make territorial gains or "sphere of influence" gains at the Empire's expense significantly in excess of what they got in OTL (Bessarabia)?
The Girays were the legitimate inheritors of the OE. Could they be installed? And which Giray descendants were available and where they did they live? The last Crimean Khan had already been deposed. Would it be possible for the Ottoman military, French and Russians to back their own member of the Giray family as a client/puppet ruler for the Empire?
Could/would Russia push hard trying to fill the vacuum using its own forces and clients (be they rebels, the Ecumenical Patriarch, a Giray they control, Phanariot Hospodars or anyone else) from 1808, and tie down much more of their manpower from 1808 onward deep in the Balkans, Caucasus, maybe even deeper into Asia Minor and the Fertile Crescent?
Might this alter Napoleon and Alexander's calculations and cancel, postpone, or accelerate the Franco-Russian war that the former started in 1812?
Could the British find themselves stretched thin between competing commitments in the eastern Mediterranean, and the emerging peninsular war?