How can I recursively find a directory by name and delete its contents (including all sub-directories and files) while keeping the directory itself?

Test run:

find . -path '*/EmptyMe/*'

Real deletion:

find . -path '*/EmptyMe/*' -delete

-path '*/EmptyMe/*' means match all items that are in a directory called EmptyMe.

One option that can be used is to nest the commands:

find . -type d -name 'EmptyMe'  -exec find {} -mindepth 1 -delete \;

The outer find -type d -name 'EmptyMe' locates the required directories, and runs the inner find command via -exec ... \;. The inner command descends into the found directory (referenced via {} ) and since we're using -delete flag here, it should follow depth-first search, removing files and then subdirectories.