Heat the oil in a medium skillet over medium heat. Add the scallions and ginger and cook until the scallions begin to soften, 1 to 2 minutes. Add the sugar, ketchup, lime juice, soy sauce, and ¼ teaspoon each salt and pepper and bring to a simmer. Cook, stirring often, until slightly thickened, 4 to 6 minutes.
Reserve half the sauce for dipping, if desired. Cook 1½ to 2 pounds of beef, pork, poultry, lamb, seafood, or tofu on a preheated grill to desired doneness. Brush on the sauce during the last 2 to 3 minutes of cooking.