Sprinkle pork with 1/2 tsp. salt, then brown in oil in a large frying pan over medium-high heat, turning as needed, 10 minutes. Transfer pork and pan juices to a 5- to 6-qt. slow-cooker. Add remaining 1 tsp. salt, the cider, Calvados, 1 tbsp. thyme, the pepper, and 1 sliced apple. Cover and cook until meat is very tender, about 4 hours on high or 7 hours on low.
Meanwhile, about 20 minutes before pork is done, heat 2 tbsp. butter in a large frying pan over medium heat. Add remaining 3 apples and cook, stirring frequently, until tender and light golden, about 10 minutes. Transfer to a bowl; tent with foil.
Transfer roast from slow-cooker to a platter and tent with foil. Strain slow-cooker juices and skim fat; set aside. Melt remaining 1 tbsp. butter in frying pan. Add flour; cook, whisking often, until golden and bubbling. Slowly whisk in juices and 1 tsp. thyme; cook until slightly thickened, 6 to 8 minutes. Transfer to a gravy boat.
Slice pork, scatter with reserved apples, and drizzle with gravy. Garnish with thyme sprigs and serve more gravy on the side.