Southern egg gravy is a gravy phenomenon in its own right. You'll rarely find it on restaurant menus, but it's deeply ingrained in Southern culinary culture.
Egg gravy is essentially the same as any white country-style gravy. Eggs generally jump in with the ingredients at the end, and therein lies the anomaly.
The eggs are hard-boiled first, thinly sliced, or diced into small, chunky bits, similar to how you'd chop them for egg salad. You could also crumble the eggs with your fingers.
The eggs are most commonly hard-boiled, but some insist on slicing the eggs instead of chopping. Some people beat the eggs first and cover the gravy base with hot milk.