Preheat grill to medium-high (or see Stovetop Variation).
Place pork in a medium bowl. Add half of the sauce mixture to the pork and gently combine without overmixing. Form into 4 patties, about ¾ inch thick. Add the remaining 1 teaspoon sesame oil and vinegar to the remaining sauce mixture and set aside.Oil the grill rack (see Tip). Grill the burgers, turning once, until an instant-read thermometer inserted in the center registers 155°F, 4 to 5 minutes per side. During the last 3 minutes of cooking, add the pineapple rings to the grill and cook, turning once, until dark grill marks appear.Combine watercress and carrot in a large bowl. Toss with 2 tablespoons of the reserved sauce. Divide the salad among 4 plates. Top each portion with a pineapple ring and a burger. Drizzle any remaining sauce on top.