Special equipment: A meat thermometer, a grill basket
Prepare an outdoor grill for medium-high heat, leaving 1 burner off (for indirect heat to finish the pork). Put the tomatoes in the center of a large sheet of foil, drizzle with 2 tablespoons of the oil, gather the edges up to the center and fold over to form a purse. Grill for 20 minutes. Remove from the heat, and, being careful of the steam, open the pouch. Pour the tomatoes and their juices into a medium bowl, mash with a fork to make a chunky sauce and stir in the vinegar, basil, 1 teaspoon of the brown sugar and 1 teaspoon salt. While the tomatoes are cooking, combine 1 tablespoon of the remaining oil, the remaining 2 teaspoons brown sugar, the paprika, cloves, 1 teaspoon salt and a few grinds of black pepper in a small bowl to make a paste. Coat the pork generously with the paste. Oil the grill grates, and grill the pork over direct heat for 6 minutes, turning occasionally. Move the pork to indirect heat, cover and cook until a thermometer inserted in the thickest part of each piece registers 145 degrees F, about 10 minutes more. Transfer the pork to a baking sheet, and let rest for 10 minutes, then cut into 1/2-inch chunks. Toss the broccoli with the remaining 1 tablespoon oil, 1/2 teaspoon salt and a few grinds of pepper in a small bowl. Place a grill basket on the grill, add the broccoli and grill, turning frequently, until the edges are charred and the insides are tender, about 6 minutes; transfer to the baking sheet. Brush the pineapple slices with oil, sprinkle with salt and pepper and grill until lightly charred, about 3 minutes on each side. Cut into 1/2-inch chunks, and transfer to the baking sheet. Brush both sides of the pitas with oil, and grill until marked, about 1 minute per side. Top the pitas with about 1/4 cup of the tomato sauce, then top with the pork, broccoli, pineapple and Cheddar, dividing evenly among them. Grill, covered, until the cheese melts, about 5 minutes.