I think Davey's got it, even the same number on the barrel. For inserting Dentocoll into molds....I still don't quite know why they needed the two plungers, or why when open one plunger would need to extend so far beyond the barrel; maybe for cleaning? I'm guessing one would put the raw stuff in the barrel, close it up for mixing, maybe add a special tip for squirting into the molds. But I can research that. Thanks, Davey! and all others who responded with their ideas. (I had thought of the giant pill-popper idea too, but didn't see why you would need the closed barrel for that.)
Found a reprint of an old article from February 1929. The hydrocoll (a hydrocolloidal modeling compound) came in sticks the size of the barrel. One would unscrew the end cap, put the stick inside, reattach the cap, and place the assembly in boiling water for four minutes. Put it vertically on a table and work the T-handled mixing rod back and forth vigorously about a dozen times. Place in cold water, and agitate a few more times. Once it is cool, remove the end cap and use the ejecting handle to eject a small amount of material into the impression tray. Smooth it, and eject some more hydrocoll to fill the tray. If a smaller amount was needed, a spare cap could be attached to reduce the diameter of the ejected material.