Preheat oven to 300°. Season roast with 1 1/2 tsp. salt and 1 tsp. pepper. Heat oil in a 4- to 5-qt. dutch oven over high heat until very hot. Brown pork on all sides, turning as needed, about 10 minutes total.
Meanwhile, cut onion, medium carrots, and celery into 2-in. chunks. Arrange around the browned roast with garlic. Pour in 2 cups cider and 1 cup broth ("I make mine from the bones of Sunday night's chicken"), adding more of either if the liquid doesn't completely cover meat. Bring to a boil. Cover with foil, pressing it down onto meat to eliminate any air gaps, then add the lid.
Braise in oven 2 to 2 1/2 hours, until a fork slides right in; "it means the connective tissue has lost the battle and the muscle is loosening." Let cool, then chill overnight. (Or let sit until cool enough to handle.)
Preheat oven to 375°. Spoon off fat from roast and discard. Transfer roast to a plate. Remove twine and break roast into 2-in. chunks with your fingers. Strain braising liquid and return to pot (discard vegetables).
Reduce braising liquid to 2 cups over medium heat, 10 to 20 minutes. Return pork to pot. Taste and add more salt or pepper if you like. Cut rainbow carrots in half lengthwise and arrange over meat. Roast, uncovered, spooning juices over carrots a couple of times, until they begin to brown and are tender, about 50 minutes. "If it isn't browning up, feel free to pop it under the broiler."
Stir in thyme. Scatter parsley on top and serve with mashed potatoes.
*Find rainbow carrots at well-stocked grocery stores and farmers' markets, or use small orange carrots.
Make ahead: Through step 2, up to 2 days.