Place the meat on the bone in a large pot, cover with water, and bring to boiling. Simmer gently for several hours (3 hrs recommended). Remove from heat until cool and refrigerate.
The next day, remove any fat that has congealed on the surface with a spoon.
Remove all of the meat from the pot and pull the meat off the bones. Discard bones. Return meat to the stock.
Add the potatoes, swede or parsnips, carrots, onion, rosemary, and peppercorns. Add salt to taste. Simmer stew until vegetables are tender—check after 20 mins so potatoes don’t overcook. Add more vegetable stock if needed.
Add the shredded leeks, mint, and chopped parsley. If you want your cawl to be thicker, you can thicken it with a flour and water paste (regular flour or substitute oat flour if gluten-free) or fine oatmeal.
Serve with Caerphilly cheese (or Gouda or Cheshire as substitutions. Sharp cheddar such as Black Bomber by Snowdonia Cheese Company can also be used.)