How to Make It
Make the carnitas. Preheat oven to 275°F. Season pork shoulder with salt and pepper. Heat oil in a large Dutch oven over moderate heat. Sear pork shoulder until golden brown all over, 5 to 6 minutes per side. Remove pork from pan. Pour off drippings and discard. Return pork to pan and squeeze juice from orange halves over pork. Add spent rinds to pan along with milk, thyme, and bay leaf. Cover with a lid or two layers of aluminum foil. Roast in preheated oven until very tender, pork is sizzling in fat, and a meat thermometer registers 195°F, 3 to 4 hours. Let pork rest for 20 minutes. Shred pork, discarding any large pieces of fat.
Make the grilled cheese: In a small bowl, fold together the blue cheese, cheddar, Parmesan, parsley, and shallot. In a second small bowl, stir together the mayonnaise and mustard.
Spread each slice of bread with the mayonnaise-mustard mixture. Top one slice with cheese mixture, 1/3 cup carnitas, and chutney, then top with remaining bread slice.
Melt butter in a cast-iron skillet or griddle over moderate heat. Press the sandwich and top with another skillet or heavy, heatproof plate. Cook until bread is golden and cheese is melted, about 5 minutes per side.