Dice ham and cook it in a large pan until tender. (It will be super convenient if this pan can also go into the oven.)
Boil eggs then slice or chop.
While the ham and the eggs are cooking make the dough. Make them exactly like you do biscuits. Cut the shortening into the flour, add the buttermilk, combine, dump onto a floured surface and cut into biscuits. Save those pieces between your cutter, we'll drop those in like dumplings.
When the ham is tender leave the liquid that cooked out of the ham in the pan, and add milk, cream of chicken soup, water, chicken stock (broth), and black pepper. Stir to combine. Add the eggs and gently still them in being careful not to break them apart. Bring to a low boil.
Drop the biscuit pieces into the boiling pan. Simmer for about 5 minutes. Turn the stove off and place the biscuits on top of the mixture.
Bake at 400°F for 25 to 30 minutes or until the biscuits are brown and the filling is bubbly.