Using a sharp knife, slice them in half lengthwise.
Cut out the white cores of the kidneys.
Soak the kidneys in a mixture of 1 teaspoon of salt and 2 cups of water for a full 12 hours in the refrigerator, rinse off, then soak in 2 cups of milk for another full 12 hours in the refrigerator.
Rinse off the kidneys, cut them into bite-size chunks, and pat them dry with a paper towel.
Heat a frying pan over medium-high heat. Add olive oil to the pan. Heat oil for 30 seconds. You want it hot, but not smoking.
Place the kidneys cut side down in the hot oil, then gently press down on them with a spatula. Sear the kidneys for 2 minutes, then turn them over and sear in the same way for another 2 minutes. Kidneys should still be pink in the middle. Take the kidneys off the heat and allow to rest on a cutting board.
Meanwhile, in the same frying pan over medium-high heat, add 1 Tablespoon of butter and diced onion and sauté for 2 minutes.
Add the broth, apple cider, thyme, and black pepper. Simmer until liquid is slightly reduced, about 4 minutes. Whisk in cider vinegar and remaining 2 Tablespoons of butter. Remove from heat, add cooked kidneys back into pan, and stir in rosemary. Plate and serve immediately.