The lime-and-soy-marinated chicken is delicious added to the coconut-curry soup, but it can also be served on its own. (Double the recipe if you want each person to have a full serving of chicken.)
Stir together 1 tablespoon lime juice, oil, soy sauce, sugar, shallot, and 3/4 teaspoon salt in an 8-inch square baking dish. Add chicken; toss to coat. Refrigerate for at least 30 minutes.
Heat broiler. Arrange chicken on top of shallot in dish. Broil 8 inches from heat source until lightly browned, about 7 minutes. Flip chicken, arranging shallots around sides. Broil until chicken is cooked through, 6 to 7 minutes more. Let stand for 5 minutes. Slice chicken crosswise. Reserve pan juices.
Transfer chicken and reserved pan juices to a bowl. Toss with basil and remaining tablespoon lime juice.
Chicken can be marinated for up to 1 day.