Prepare brine. Heat 2 cups of the water in a saucepan over medium-high heat. Add salt, sugar, granulated garlic and pepper and stir until dissolved. Add to remaining ice water and stir in vinegar. Place roast in a heavy-duty plastic bag, pour brine over and seal bag at opening to immerse roast in brine. Place in a container, and refrigerate for 6 to 12 hours.
Remove roast from brine. Discard brine. Rinse roast, pat dry with paper or clean cloth towels. Rub with olive oil, and season with salt and pepper.
Preheat oven to 325 degrees. Heat 2 tablespoons olive oil in a large skillet (or Dutch oven) over medium-high heat. Add roast, and brown well on all sides. The browning will enhance flavor!
Transfer meat to a well-greased roasting pan (or leave in Dutch oven). Stir cider and stock together, and pour enough in pan to cover ½ to ¾ inch. Add apples, onion, garlic and bay leaves to pan. Cover tightly with heavy foil or lid, and place in the preheated oven.
Check oven after 2 hours to make sure there is adequate liquid in the pan. Add as necessary to keep at least ½ inch of liquid at all times. (Note: if you got distracted and allowed the liquid to evaporate, just deglaze the pan with additional cider and broth.) Depending on the size of the roast, it will take at least 5 hours for the braise. Check every hour or so after the first 2 hours, and continue until meat pulls away from the bone easily.
