Preheat oven to 325 degrees F. In a medium saucepan bring broth to boiling. Slowly add grits, stirring constantly. Gradually stir about 1/2 cup of the hot mixture into the egg. Return egg mixture to the remaining hot mixture in saucepan; stir to combine. Remove saucepan from heat. Add cheese, green onion, and butter to grits, stirring until cheese and butter melt.
Pour grits mixture into an ungreased 1-quart casserole. Bake, uncovered, for 30 to 45 minutes or until a knife inserted near the center comes out clean. Let stand for 5 minutes before serving. If desired, top with cilantro.