Creating Overlapping Donuts

I have one shapefile of larger overlapping buffers in which I need to remove another shapefile of smaller overlapping buffers. These shapefiles have unique identifiers that link the records of one shapefile to the other. My intention is to create one shapefile of "donuts" for each of these records based on this unique identifier.

Using the erase tool in this instance won't work since the polygons are overlapping. My wish is not to have multiple donut holes in the larger overlapping buffer layer. Also, the process seems very time consuming to complete in an editing session since there are a couple thousand unique donuts that need to be created.

Any suggestions? Thanks!