For the pork: Score the "fatty" side of the pork with a sharp knife just deep enough for the marinade to reach the meat. Mix the oil, garlic, smoked paprika and salt into a paste and rub liberally around all sides of the pork shoulder. Top with the thyme and shallots. Refrigerate for 4 hours or overnight. Preheat the oven to 400 degrees F.
Place pork in a roasting pan on top of a rack. Add 1 cup cold water to pan and roast for 30 minutes uncovered.
Check the pork liquid after 30 minutes, if no water remains add a bit more, then cover the meat with parchment paper followed by aluminum foil. Reduce the oven temperature to 300 degrees F, and return the pork to the oven until the meat is tender, 1 hour 30 minutes, and then cool uncovered to room temperature.
While wearing gloves, carefully run your fingers underneath the top "fatty" layer of pork, and try to remove the fat in 1 or 2 pieces and set aside.
Gently shred the pork meat off the bone with a fork or by pulling with your fingers, use gloves as pork will be warm closer to the bones, and set in a large bowl. Using a sharp knife, finely dice the reserved pork fat and fold into the pulled pork meat.
For the pickled vegetables: Meanwhile, whisk together the vinegar, salt, sugar and pepper flakes. Pour over the carrots, cucumber and jicama and refrigerate for 2 hours.
For the aioli: Mix together the mayonnaise, capers, cornichons, tarragon, lemon juice and mustard, if using, in a medium bowl. Season with salt and refrigerate until ready to use.
For assembly: Cut the buns in half and lightly toast or grill. Spread some aioli on each half. Place a small amount of pickled vegetables on the bottom half of each bun, then place 2 to 3 cilantro leaves on top. Add 2 to 3 ounces of roasted pork and as much sriracha sauce as you like, and secure the top half of the bun with a toothpick.