In top of double boiler, combine 2 egg whites, nutmeg, salt, vanilla, sugar, coconut, milk and butter. Stir over hot (not boiling) water 5 minutes, or until mixture has slightly thickened (do not overcook). Remove from heat; cool to room temperature.
Beat remaining 2 egg whites only until they stand in soft, stiff peaks (do not beat too stiff). Fold into cooled coconut mixture. Empty into pie shell.
Bake in very hot oven (450°F.)10 minutes; reduce heat to moderate (350°F.) and bake 30 to 40 minutes longer, or until filling is firm in center. Serve cold.