Trouble setting up recursive particle systems

I have a donut (from following the Blender Guru tutorial), which uses a particle system to instance sprinkles, and I am trying to create raining donuts with a plane with an emitter particle system, but the instanced donuts do not have any sprinkles on them.
donut_rain.blend.xz (960.8 KB)

Is it possible to set up recursive particle systems in Blender?