Combine butter, onion, rice, broth, and 3/4 cup water in a 2-quart microwave-safe dish. Cover with heavy-duty plastic wrap, and vent. Microwave at HIGH 15 minutes, rotating dish a half-turn after 7 1/2 minutes. Stir in remaining 3/4 cup water; cover and microwave at HIGH 5 minutes.
Stir green chiles and chicken into rice; cover and let stand 5 minutes. Uncover and stir in cheese, 2 tablespoons water, salt, and cilantro.