Remove pork from packaging and pat dry with paper towels.
Place fat side down on a rimmed baking sheet or cutting board.
Score the pork shoulder by making 1 inch deep slices across the entire ham about 1 inch apart. Turn the ham and make slices in the other direction, creating a crosshatch pattern.
Lightly coat the pork with olive oil and sprinkle with dry rub. Set aside while you prepare the grill.
Using lump charcoal, prepare a fire in the Big Green Egg. We use a custom Eggspander to raise the grill grate above the fire, adding hickory for hardwood smoke flavor.
When the fire has cleaned up and only producing light colored smoke, place the pork butt onto the grill grate and cook with the grill closed for 5 hours, checking about every hour, maintaining a temperature of 250°- 300°.
Drizzle pork with bbq sauce and continue to cook about another hour or until the pork reaches 200°.
Remove from the grill to a rimmed baking sheet or cutting board. Rest for 20 minutes before shredding.
Pulled pork can be served immediately or if you are making it ahead, cool slightly and store in a zip top bag or air tight container up to 3 days.
For Sandwiches: Lightly toast the buns and top with coleslaw. Add pulled pork and top with your favorite sauce and the top bun.