Preheat smoker to 275 degrees Fahrenheit (F) using apple or cherry wood.
Trim excess fat off of pork belly so it is uniform in shape and no excess fat or meat is hanging off the belly. Place pork belly fat/skin side down on the board so flesh is facing you.
Make 1-inch vertical slices in the pork belly, taking care not to cut all the way through. Cut about 2/3's through the flesh.
Coat both sides of the pork belly with olive oil and the dry rub. Rub the dry rub into the ridges created by the vertical slices.
Place into the smoker and smoke for 2 hours or until the internal temperature of the pork reaches 200 degrees F using an instant read thermometer. When the probe goes into the belly it should slide in easily. That is when you remove from the smoker.
Prepare the grill for direct grilling, targeting 450 - 500 degrees F over the flame. Place the pork belly over direct heat and grill each side for 2 minutes. Be sure to close the lid between flipping to avoid too much air causing flare ups, the fat from the belly will drip into the flame.
Remove the belly from the grill and finish slicing the vertical slices. Apply sauce and serve as small long strips for a meal, or cut into cubes for an appetizer.
For the Sauce
In a medium size bowl, combine Hoisin sauce, Thai chili sauce, lime juice, and graded fresh ginger. Stir to incorporate.