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.
Mix the paprika, garlic power, brown sugar, dry mustard, and salt together in a small bowl. Rub the spice blend all over the pork. Cover and refrigerate for at least 1 hour, or up to overnight.
Preheat the oven to 300 degrees F. Put the pork in a roasting pan and roast it for about 6 hours. An instant-read thermometer stuck into the thickest part of the pork should register 170 degrees F, but basically, what you want to do is to roast it until it's falling apart.
While the pork is roasting, make the barbecue sauce. Combine the vinegar, mustard, ketchup, brown sugar, garlic, salt, cayenne, and black pepper in a saucepan over medium heat. Simmer gently, stirring, for 10 minutes until the sugar dissolves. Take it off the heat and let it sit until you're ready for it.
When the pork is done, take it out of the oven and put it on a large platter. Allow the meat to rest for about 10 minutes. While it's resting, deglaze the pan over medium heat with 3/4 cup water, scraping with a wooden spoon to pick up all of the browned bits. Reduce by about half. Pour that into the saucepan with the sauce and cook 5 minutes.
While the pork is still warm, you want to "pull" the meat: Grab 2 forks. Using 1 to steady the meat, use the other to "pull" shreds of meat off the roast. Put the shredded pork in a bowl and pour half of the sauce over. Stir it all up well so that the pork is coated with the sauce.
To serve, spoon the pulled pork mixture onto the bottom half of each hamburger bun, and top with some slaw. Serve with pickle spears and the remaining sauce on the side.
Combine the cabbage, carrots, red onion, green onions, and chile in a large bowl. In another bowl, stir together the mayonnaise, mustard, vinegar, lemon juice, and sugar. Pour the dressing over the cabbage mixture and toss gently to mix. Season the cole slaw with the celery seed, hot sauce, salt, and black pepper. Chill for 2 hours in the refrigerator before serving.