Entity Reference | View - Display content based on a entity reference field's value that is randomized?
you just have to configure a view that:

shows 1 breed, sorted random.
has another display that shows dogs based on a contextual filter on the breed nid
Install views field view
use the block from step 2 as a field in your first display and give the nid of the breed as a contextual filter to your views field.

So this views will show one random breed and has a view field to show the dogs that are of that breed. Hope this helps you out.