The way I did it successfuly was first applying an audio generator to the LCD shutter and varying the amplitude and frequency to find the optimal darkening setting for that particular shutter, within reasonable amplitude limits. It was around 90Hz and 15 Volts p-p.
Then built an oscillator at those exact frequency and amplitude, switched-on by a phototransistor, or a plain on/off switch if you want it manual.
Miguel