Okay, WHAT is going on with my brakes??!

Cardone = crap.
Checker = crap.
"Remanufactured" = crap.

This what you're experiencing is typical of a master cylinder with leaky seals: when you hit the brake pedal hard, the seals flare and make a decent seal. When you slowly apply the pedal, the seals don't flare and fluid runs past them.

Put on a new master cylinder, Raybestos #MC36426. Make sure you bench-bleed it thoroughly before you install it. Your trouble will go away.