Step 1

Preheat the oven to 350°. Arrange the garlic cut side up on a sheet of foil, drizzle with oil and season with salt and pepper. Wrap the garlic in the foil and roast for about 1 hour, until very soft. Let cool, then squeeze the garlic cloves out of the skins into a medium bowl. Add the thyme, mustard powder, paprika, ginger, orange zest and the 1/4 cup of oil and mash into a paste.

Step 2

Reduce the oven temperature to 300°. Set a rack in a flameproof medium roasting pan and put the pork on it. Season the pork with salt and pepper, then rub it all over with the garlic paste. Cover with foil and roast for 6 hours, until an instant-read thermometer inserted in the thickest part of the meat registers 200°; uncover for the last 45 minutes of cooking. Transfer to a work surface and let cool, then shred. Discard the fat and bones.

Step 3

Spoon off all but 2 tablespoons of fat from the roasting pan. Add the sugar, both vinegars and 1/2 cup of water to the pan and cook over moderately high heat, whisking, until bubbling and the sugar dissolves, 5 minutes. Add the pork and toss to coat. Season the pork generously with salt and pepper and toss again; keep warm.

Step 4

In a saucepan, whisk the ketchup, cola, vinegar, hot sauce, molasses and cornstarch and bring to a boil. Simmer over moderate heat, stirring, until glossy and thick, 8 minutes; keep warm.

Step 5

Pile the pork on the buns and top with the barbecue sauce and coleslaw. Close the sandwiches and serve, passing the habanero vinegar at the table.