Preheat your oven to 325 F.
Season the roast with salt and pepper and heat the oil in a large skillet. When the oil is hot, brown the roast on all sides. About 2-3 minutes per side.
While the meat is browning, prepare the veggies. Peel and cut the potatoes into chunks, peel and cut the carrots into 2-inch lengths, and peel the shallots and garlic cloves.
Toss the vegetables with the remaining oil and season with salt and pepper.
Lay the vegetables and thyme sprigs in the bottom of the baking dish and set the roast in the middle of the vegetables. Add 1 cup of chicken broth and 1 cup of apple juice to the bottom of the pan.
Roast the pork at 325-F for about 45 minutes or until the roast reaches an internal temperature of 101-105 F. Remove the vegetables from the bottom of the pan and transfer them to a baking sheet.
Add the cut apples to the bottom of the baking dish and arrange them around the pork. Continue baking until the pork reaches 145-F, then remove it to a cutting board to rest. Scoop the apples out of the roasting pan and set them in a bowl to keep warm.
When the roast is out of the oven, turn the broiler to high and broil the vegetables for 5-6 minutes or until they're browned. Tuck the shallots down, or they'll burn when broiling.
Make the gravy by mixing the flour with the remaining 1 cup chicken broth then add this to the juices at the bottom of the baking dish.
Turn the heat to low and mix until the gravy is simmering and thickened. Add the apple cider vinegar and grainy mustard, then adjust the seasoning. Simmer for 2-3 minutes then transfer it to a gravy boat.
Slice the roast, cutting evenly between the bones. Transfer any juices to the gravy. Serve along with the apples, roasted vegetables, and gravy.