The glasses have been disassembled and a basic means of triggering the LCD shutters has been determined.

Each LCD lens has a small mylar ribbon connector going to the main PCB.  There is one at each corner.

For each connector, the inner contact (nearest the nosepiece) is "+" and the outer contact (nearest the lenses) is "-".

A positive voltage through a 1k resistor (with ground on the other contact) was observed to trigger the LCD shutter on the matching side. A resistor was used for some basic current limiting - a minimal nod towards protecting the hardware from something going amiss.

3V was sunglasses-dark, 5V was nearly opaque. Reducing the resistor value had some effect on the observed darkness of the triggered lens, but not as much of an effect as changing the voltage. The LCD triggering appears to be primarily voltage-controlled.

A brief experiment using PWM to drive the trigger signal yielded interesting results.  Using an Arduino, a lens was triggered with a 50% duty cycle [via analogWrite(lcdShutter,127) ] and the lens was observed to be darkened less, and the tint appeared even and stable. Voltage level has an effect on LCD lens darkness, but PWM appears to be a much better way to control "tint level".

Of course, the Peril-Sensitive Sunglasses have no need of half-measures so we won't be needing that, but it's nice to know.

This gives us enough to work with.  We know how to control the shutters now.