Whisk together peanut butter, soy sauce, lime juice, brown sugar, chili-garlic sauce, and ginger in a small bowl. Whisk in water, 1 Tbs. at a time, until spreadable but still thick.
Preheat a grill to medium-high (400°F to 450°F). Use hands to mix pork, breadcrumbs, green onions, soy sauce, sesame oil, ginger, and garlic in a medium bowl until blended; shape into 4 patties, about 4 in. in diameter and 1/2 in. thick. Place burgers on oiled grates; grill, uncovered, until well browned and a thermometer inserted in center of a burger registers 160°F, about 5 minutes per side. During last minute of cooking, add rolls to grill, cut sides down; toast lightly.