Got it my first time killing Kael’thas.
The Polar Bear from the Wrath dailies. I only wanted to do the daily for money, and then it showed up. I was new to the game, so I thought it wasn’t a big deal.
Honorable mention goes to Ashes of Al’ar and Ji-Kun mount in the same day.
Define 1st run.
I got WOD HFC mount on 1 chars first run.
However she was the newest of many wod farmers. So I had a few weeks there on many other chars.
Technically a first run drop, maybe, if really loose with the definition.
I got Thundering Onyx Cloud Serpent on my 2nd attempt, and Headless Horsemens mount on 1st.
Though I’ve done my time. Invincible took 211 runs and I’m still hunting for Aeonax after 6 years.
I got the full Hand of Bahmethra the first week of patch 9.1 when I stumbled upon a group killing a tormentor in the Maw. I didn’t even know what tormentors were at that point.
First try, got the Onyxia mount. Also happened for the first time I ran King’s Rest, that raptor is doooope. And then new Kara in Legion, first run I did the mount dropped off the first boss and my raid leader gave it to me because he thought it was “ugly.” Maybe doesn’t count, but pretty rad nonetheless.
Back in Wrath, my realm made a pug group for 25 OS +3D. Took us a few attempts, but we managed to push through and we all saw that beautiful Twilight Drake up for roll. I see a few rolls in the 30’s, a 20 or two, then the rest of the list being high 80’s - 90’s, and myself rolling a 100 after everyone had hit “need” for it making me one of the only few to have the drake on my realm.
First Wrath epic came off of that boss too before getting the mount. Good old Hyaline Helm of the Sniper with hit/haste.
I remember people getting hit by lava wall… a lot. One of my favorite raids for certain. 
Wish I could say I got them first run. I did get a string of lucky mount drops couple weeks ago (all in one night, might i add)
Got Midnight from Legion Kara. (Yes, Midnyte riding Midnight 
https://i.imgur.com/6UBxXnK.jpg)-been farming this since it came out
Got Felsteel Annihilator from HFC
Got Venthyr mount from Korthia
Got Kyrian mount from Korthia
I got my brewfest kodo on my classic character the first time I went in there, after god knows how many years of not being able to get it on retail.