In a 6-quart or larger slow cooker, whisk together apple cider, 1/3 cup apple butter, ground mustard, flour, and garlic.
Pat pork butt dry with paper towels. Combine the salt, brown sugar, and pepper in a small bowl. Rub mixture all over pork.
Place pork fat side down in the slow cooker. Scatter onion and thyme sprigs around pork.
Cover and cook on LOW for 6 to 7 hours or on HIGH for 8 to 9 hours.
Remove pork to a carving board. Tent it with foil and let sit 15 to 20 minutes.
Spoon the fat off the surface of the liquid in the slow cooker. Or, if you have a fat separator, pour it into the fat separator and discard the fat. Then return the remaining liquid to the slow cooker.Discard the thyme sprigs. You can strain out the onion slices or leave them.
Whisk the remaining 1/3 cup apple butter, vinegar, and cayenne pepper into the slow cooker. Season with salt and pepper to taste.
Use 2 forks to separate the pork into large chunks. Pour gravy over pork and serve.
Notes
Note, if you want the gravy thicker, transfer it to a saucepan. Mix 2 to 3 teaspoons of cornstarch with an equal amount of cold water. Add to saucepan. Bring to a boil and simmer for a few minutes.