Brain Surgeon (Mobility Kills)

is one of the many challenges in Titanfall.

How to Get
This challenge is pretty easy to do. To rodeo, just jump on an enemy Titan, and you should auto mount. Be aware that the enemy pilot can hop out and kill you while rodeo'ing. Please note that dooming Titans will not count towards this challenge. You must rodeo and do damage until the pilot ejects or the Titan is destroyed.

Both Auto-Titans and piloted Titans count towards this challenge.

Spitfire LMG with Slammer Mod is really helpful