Sounds exactly like a warped rotor. Had that same issue on several vehicles over the years. Usually comes from too hard a stop from too fast a speed. Even anti-lock brakes have their limits. The rotors can actually heat up so quickly that the metal will warp causing shimmy and/or pulling to one side (the brakes will now be out of balance with each other) as well as pedal pulsations. So far managed to avoid that this time around.
"There's gold in them hills" (and I'm still stuck in Florida!) ...well, in south Miss now, lol
Black '07 Compass Sport 5-speed 4X4 ("Fortuneer"), K&N filter, cat-back exhaust; RR lift, sunroof, trailer tow group, Boston Acoustic audio, sport bars, rear taillight guards, more to come... Finally got it back from the wife so I can let my Neon R/T rest for a rebuild.