The first step was to Reverse Engineer the paper cutter board and make a schematic in KiCad. Armed with the device datasheet (thanks, Google!), I attacked the board with a multimeter and some patience.
The writing on the small resistors is too small to read (except for the 200R and 0R36), so the measurements in the schematic reflect what they measure in-circuit. They're 0402-sized components, so I'm not likely to try to change them out anyway. The capacitors are a similar story; I'm sure only of the value of the big electrolytic.
Later, I'll hook the circuit up to an Arduino and see if I can make it cut paper!