Whisk chicken broth, oyster sauce, soy sauce, fish sauce, white sugar, and brown sugar together in a bowl until well blended.
Heat large skillet over high heat. Drizzle in oil. Add chicken and stir fry until it loses its raw color, 2 to 3 minutes. Stir in shallots, garlic, and sliced chilies. Continue cooking on high heat until some of the juices start to caramelize in the bottom of the pan, about 2 or 3 more minutes. Add about a tablespoon of the sauce mixture to the skillet; cook and stir until sauce begins to caramelize, about 1 minute.
Pour in the rest of the sauce. Cook and stir until sauce has deglazed the bottom of the pan. Continue to cook until sauce glazes onto the meat, 1 or 2 more minutes. Remove from heat.
Stir in basil. Cook and stir until basil is wilted, about 20 seconds. Serve with rice.