1. In a heavy saucepan saute the onions in the butter until tender but not brown. Add the flour, salt and pepper and blend well. Gradually add the milk and cook, stirring, until thickened.
2. Add the cheese, stir until melted and add the eggs. Reheat without boiling and serve on rice or toast or turn the mixture into a shallow six-cup casserole, dot with additional butter and broil until lightly browned.