This looks to me like a linear pneumatic actuator.
Pneumatic cylinders can be very fast acting, depending on the air pressure on hand, but they can be hard to control at any position other than their end stops, so they are usually used to switch between a limited number of fixed positions. Although the actuators themselves can be small, they require significant infrastructure to run (air compressors, filters, pipes, valves, pressure gauges & limiters etc.).
Pneumatic cylinders are typically single or double acting:
Simple pneumatic actuators have valves which direct either atmospheric pressure, or pressurised air into the cylinder, so you might have one pneumatic solenoid valve for a single acting cylinder, while for a double acting cylinder you might have two pneumatic solenoid valves or a single three-way value depending on whether you want to pre-exhaust for faster actuation.
To answer the other part of your question. This system basically performs the Tabelcloth Trick with a burger bun. All you really need is for the mass of the bun, and thus it's inertia, to be high enough that it stays in place against the friction of the mechanism which hold it, for the time it takes for that mechanism to get out of the way.
The higher the jolt (or jerk) of the move, the more likely it is that the kinetic friction will dominate over the (usually higer) stiction (static friction), and a pneumatic actuator with no flow limiter has the potential to have quite a high jolt coefficient.