Two flavors of weapon that are specially regulated by the National Firearms Act of 1934. "Title I" firearms are things like normal rifles, shotguns, and handguns, such as can be sold by any gun dealer, or face-to-face between citizens of a state.
"Title II" of that law covers things like machine guns, silencers, short barreled rifles (SBR), short barreled shotguns (SBS), destructive devices (grenades, explosive munitions, AND large-bore weapons like 20mm rifles), and a catch-all category of odd concealable firearms known a "Any Other Weapons" (AOW).
Short-Barreled Shotguns are any firearm meeting the definition of a title I shotgun (to include being designed to fire from the shoulder) which have a barrel less than 18" long (and/or 26" overall). If you start with a shotgun and cut it down to shorter than 26" overall, and/or less than an 18" barrel, you are making a SBS.
One of the categories listed under Any Other Weapons are smoothbore handguns. If you start with a bare receiver and build a shotgun that is less than 26" overall and/or less than 18" barrel length -- but it has only a pistol grip, no butt-stock -- you would be making one kind of AOW. If you really want to do this, rather than make an SBS, you can start with one version of "shotgun" that came from the factory with only a pistol grip. That's usually easier than finding a factory-new bare receiver.
(The ATF has ruled that when a weapon is made, ONCE, into a shoulder-fired weapon, it is either a rifle or shotgun. That means that anything else you make from it that is in pistol configuration is "made from" a shotgun or pistol, which is one of the definitions of an SBR or SBS under the NFA. This is why you can't make a Thompson Contender that is in RIFLE configuration back into a pistol without registering it as an SBR.)
If you're going to build a shorty shotgun of some sort, you'll pay the same price to make and register either an AOW or SBS. If you register it as an SBS, you can run it with a stock or without. If you register it as an AOW, you can't add a stock. Might as well go SBS.