Food Network teaches how to crush, slice and mince garlic. Peel off some of the papery skin from the garlic and then smash the head of garlic with the heel of your hand to loosen the cloves; if you only need a few cloves, leave the head intact and pull some off. Separate the cloves. To peel a clove, cut away the root end with your knife. Lay the flat side of the knife over the clove while holding the knife handle, then with the heel of your free hand carefully whack the knife against the garlic to separate the skin from the clove. To crush the peeled garlic, lay the flat side of the knife over the clove and smash it again. To slice peeled garlic, lay the clove flat on the cutting board and hold it with the fingertips of one hand, keeping them curled under. Using a rocking motion with the knife, make thin slices by moving the knife slowly across the clove. To mince peeled garlic, lay the flat side of a knife over the clove and smash it. Roughly chop the clove then move your free hand flat across the tip of the knife and use a rocking motion to chop the garlic until it’s finely minced.
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.