Eggs are used as a thickener for sauces, but they only work if they are added slowly. The term temper refers to temperature, and it is a method that gently raises the temperature of the delicate egg so that they can be added to a hot liquid without scrambling.
To temper, a small amount of hot liquid is added and quickly stirred into the eggs. The warmed egg is then added and quickly stirred into the pot of hot liquid. Cooking continues, all the while stirring, until the desired thickness is reached.
The look of “desired thickness” is the tricky part. If you stop cooking too soon, the sauce will be watery. Too long on the heat will scramble the eggs, no matter how carefully you tempered. The method for checking doneness is to dip in a wooden spoon and run your finger across the back of it. Your finger should leave a definite trail through the sauce that does not fill back in.
Cool quickly to stop the cooking, as scrambling will still take place in a hot pot even if the flame is off. This is best done over an ice bath.