Comparing to other EU4 achievements, Baa Baa Black Sheep is a relatively easy to be done. But the starting isn’t too easy, as Qara Qoyunlu starts with less than 30% of religious unity and the non-true faith religions are either Sunni or Coptic. Their regions have -2 local missionary strengths to make them harder to convert. The unrest base in the Armenian mountains will make the rebels more likely to spawn. In the same time, powerful and unfriendly neighbors like Mamluks, Ottoman, and possibly Persia are looking forward to take the rich Mesopotamia. Such starting can be intimidating for inexperienced players.
The good news is, every problem in EU4 has a solution, every nation can be a powerful empire. Most of the heretical regions are belong to the Armenian and Mashriqi culture. As soon as these cultures are accepted and there’s a positive piety, converting will be fairly easy. Also, raising autonomy will slow down the spawning rebels.
As Qara Qoyunlu is one of the few Shia nation, raising piety will be easy. Aq Qoyunlu, Georgia, Turkish minors, and most of the Arabian nations can be easy targets. Attack the Timurids when it is in war or under explosion. In the same time, make Tabarestan and Shirvan to transfer trades to you because Tabarestan owns the Persian trade center.
Also, beware of the Ottoman/Mamluks because it always wanted to expand into Mesopotamia and they are stronger than Qara Qoyunlu in the starting. Scorched earth and build forts in deserts if necessary. Luckily, Mamluks is my ally in this game and we are successfully blocking the Ottoman expanding into the East.
Second war on Timurids when Mamluks and manpowers are ready
Tips: Occupying Timurid provinces will decrease their wool production and thus lower their ranking. So it is possible to get the achievement easily if Qara Qoyunlu can survive the beginning and winning a war against Timurids.
PS: I am going for the Georgia in my mind achievement after this because I have one Georgia now, and took the exploration idea as the second idea.