Prepare a medium-hot fire in a charcoal grill with a cover, or preheat a gas grill to medium high.
For the Dressing: combine the mayonnaise, basil, cilantro, green onions, and lime juice in a bowl. Cover and refrigerate.
For the Patties: combine the fish sauce, jaggery, and chili sauce in a medium bowl. Add the pork, peanut butter, ginger, garlic, and star anise and blend loosely with a fork. Form into 6 equal patties the approximate dimensions of the rolls, making a slight depression in the middle of the patties to compensate for the tendency to bulge during cooking.
Brush the grill rack with oil. Grill the patties with the grill top closed for 4 minutes. Turn and grill until an instant-read thermometer inserted into the center of the patties registers 160 degrees F, about 4 minutes longer.
During the last 2 minutes of grilling the patties, place the rolls, cut side down, on the grill to toast.
To assemble the burgers: spread the roll bottoms with a thin layer of dressing, followed by a lettuce leaf. Add the patties and top with generous dollops of the dressing. Add the roll tops.