I have an '04 GT w/ some modifications, including a full exhaust. My first guess was an exhaust leak and I'm honestly hoping it's just that and not something to do with my transmission or, god forbid, the motor.

I provided a sound clip of it because I'm terrible at describing sounds.

Any help is appreciated.

 

That sounds like piston slap or valve knock (not rod knock). Valve knock could is usually a lifter that needs adjusting or in the case of your engine (it's hydraulic) a failing hydraulic lash adjuster..

https://mustangforums.com/forum/4-6l...placement.html