•3 tablespoons all-purpose flour, plus more as needed to thicken sauce
•2 tablespoons plus 1/2 teaspoon kosher or sea salt
•2 teaspoons freshly ground black pepper
•2 tablespoons olive oil or other vegetable oil
•1 3-1/2- to 4-1/2-pound boneless pork shoulder (pork butt or Boston butt), excess fat removed
•1 pound whole pearl onions (frozen and thawed or fresh)
•10 ounces sliced button (white) mushrooms
•1 teaspoon ground allspice
•5 sprigs fresh thyme
•2 cups hard cider, preferably local
•1-1/2 cups reduced-sodium chicken broth
•3 large apples, any variety, skin on, cored and cut into 1/2-inch cubes
•Garnish: fresh thyme sprigs
In a shallow bowl, stir together 3 tablespoons of flour, 2 tablespoons of salt, and the pepper. Roll the pork shoulder in the flour mixture to coat; then shake off any excess.
Preheat your oven to 325° and set a rack to the middle position. On the stovetop, set a large Dutch oven over medium-high heat. Add the oil and let it heat for a minute; then add the pork. Brown the meat all over, 3 to 4 minutes per side, adjusting the heat as needed to keep the flour from burning. Remove the pork and all but 3 tablespoons of fat from the pot. Set the pork aside and discard the excess fat.
Add the onions to the pot and increase the heat to high. Cook, stirring often, until browned in parts, 3 to 5 minutes. Remove half the onions with a slotted spoon and set them aside. Add the mushrooms to the remaining onions and cook, stirring often, until the mushrooms are nicely browned and have visibly shrunk, 3 to 5 minutes. Add the allspice and thyme and stir; then add the cider, stirring the bottom of the pot to pick up any browned bits. Add the chicken broth, and then return the pork to the pot. Bring the liquid to a simmer; then cover the pot and transfer it to the preheated oven. Cook 2 hours; then remove from the oven.
Add the apples and reserved onions to the pot, stir, and return to the oven. Cook 30 minutes more. The pork should now be very tender when poked with a fork; if not, cook 15 minutes more. Remove the pork from the pot, transfer to a carving board, and tent with foil. Using a slotted spoon, transfer the apples and onions to your serving bowl.
If you like, thicken the sauce: Pour about cup of it into a small bowl and whisk in 2 tablespoons of flour until smooth; then stir it back into the pot and simmer until the rest of the sauce thickens. Check the seasoning and add more salt if needed.
Slice or shred the pork as you prefer; then transfer to your serving bowl with the onions and apples. Pour the sauce over all, garnish with additional fresh thyme, and serve over polenta, noodles, mashed potatoes, spaetzle, or couscous