For the rub:
Combine 1 Tbsp (3 tsp) sugar, 1 Tbsp (3 tsp) dried basil, 1 Tbsp (3 tsp) dried oregano, 1 Tbsp (3 tsp) salt, and garlic powder in a small bowl. Evenly rub the mixture over the pork to coat. Let stand at room temperature for 45 minutes.
Set up grill for medium-low indirect heat cooking (275°F- 300°F). Place an aluminum pan, half filled with water, on the charcoal grate under where the pork will be.
Place the wood chunk on the lit charcoal. Set the pork, fat side up, on the grill grate over the aluminum pan and close the lid.
Grill, adding enough charcoal to maintain temperature after each hour, until an instant read thermometer inserted into the center of the pork without touching the bone registers 160°F, about 3 1/2 to 4 hours.
Wrap the pork in a large sheet of heavy-duty aluminum foil and return to the grill over indirect heat. Close the lid and cook until the internal temperature reaches 195°F about 1-2 hours longer. Remove from the grill and let rest in foil for 30 minutes.
Meanwhile, heat the oil in a medium nonstick saucepan over medium. Add the onion, fresh minced garlic, 1 tsp dried basil, and 1 tsp dried oregano; cook, stirring occasionally, until slightly softened, 3-4 minutes.
Stir in the crushed tomatoes, tomato paste and 2 tsp sugar; bring to a boil, reduce heat to medium-low and simmer, stirring occasionally, until thickened, about 20-22 minutes. Remove from the heat and stir in 1/2 tsp salt, and black pepper.
Unwrap the roast and if cool enough to handle, shred the meat, discarding any large pieces of fat. Transfer pork to a bowl and stir in the sauce until well mixed.
Top slider rolls evenly with pork and sprinkle generously with shredded Parmigiano-Reggiano cheese.
Recipe and Picture from Marge Perry & David Bonom at www.anolon.com