In a food processor fitted with a steel blade or in a blender, combine the basil, parsley, and pine nuts. Add olive oil and puree until smooth. Add cheese and stir or pulse. Place pork chops in a zippered plastic bag or shallow glass dish. Spread half of the pesto on both sides of the pork chops; reserve the rest. Sprinkle with salt and pepper. Seal the bag or cover and marinate in the refrigerator for 15 minutes.

Prepare a grill for direct heat cooking: Light a charcoal grill and let the coals burn until covered with light grey ash. Spread the coals. Or, light a gas grill according to manufacturer’s directions. Place the cooking grate on top and let it heat for a minute or two until hot.

Meanwhile, cut the potatoes into 1/2x1/2x2-inch sticks or logs. Sprinkle the potato logs and tomatoes with salt and pepper and drizzle with olive oil.

Place the potatoes and tomatoes, skin-side down, in the center of the cooking grate; grill 20 minutes, or until tender, turning occasionally.

Bring the coals together in a concentrated area. Sear the pork chops over direct heat for 2 to 4 minutes, rotating once to create criss-cross grill marks. Turn and move to the side of the coals. Cover and continue grilling by indirect heat for 30 minutes for medium (160 F/71 C) or 45 minutes for well-done (170 F/77 C).

To serve: Arrange potato sticks from one potato in a criss-cross pattern on a warmed serving plate to make a log box; place two tomato halves in the center of the box. Repeat with the remaining potato and tomato on a second plate. Add a chop to each plate and drizzle with the reserved pesto.