This is a problem particular to the 2012 Ford F-150 with V-6 Ecoboost engine.
Right after starting the engine, stepping on the brake, putting it in drive or reverse, do you hear a grinding or buzzing noise coming from the engine compartment?
Near the driver's side front bumper?
If you open the hood, it sounds like it's coming from under the air filter?
It's the vacuum pump.
Seems there is not enough vacuum from the turbocharged engine to effectively operate the master cylinder booster, so they added a supplemental pump.
Later models come equipped with a mechanical pump.
Motorcraft Part # BRPV10
I have a video. Look and listen. You'll hear the noise shortly after the engine starts up.
Price for replacement vacuum pump from Ford dealer is over $600.
Parts were available from RockAuto or Amazon for half price, but they have become scarce for some reason.
It's very easy to replace. The pump is directly behind the headlamp assembly.
Remove the headlamp assembly and the pump is easily accessible.
Power connector, vacuum hose, and a couple of bolts.