Place the pork, fat side up, in a roasting pan fitted with a rack insert. Place the salt, pepper, brown sugar, paprika, thyme leaves, garlic, vinegar, and cayenne in a food processor and pulse until well combined. Add extra-virgin olive oil until you have a nice paste. Rub all over the pork, being sure to get into the nooks so the salt can penetrate the meat and pull out the moisture - this will help form a crust on the outside when cooked. Cover the pork with plastic wrap and marinate in the refrigerator for at least 3 hours or up to overnight.
Allow the meat to sit at room temperature for 30 minutes before cooking. Preheat the oven to 325 degrees F.
Roast the pork for 3 1/2 hours, uncovered, until the outside is crispy-brown (it should look like mahogany). Let the meat rest on a cutting board for 10 minutes before slicing.
While the pork is cooking, prepare the slaw. Combine the mustard, mayonnaise, sour cream, lemon juice, vinegar and sugar. Mix well and add finely sliced savoy cabbage, purple cabbage, green onions and carrots. Season with salt and pepper, to taste, cover with plastic wrap and set aside in the refrigerator until you are ready to serve.
To serve, cut the brioche buns in half and pile up some sliced pork. Top with a large spoonful of coleslaw and place the top half of the bun on top. Garnish with parsley.