In a wok or frying pan, heat enough cooking oil to reach a depth of at least three inches. The temperature should be before it starts to smoke. Sorry, I don’t use kitchen thermometers. You can test the temperature by dropping a small piece of bread into the hot oil. If the bread browns too fast, the oil is too hot.
Make three club sandwiches: Spread 1/4 c. of chicken salad on a slice of bread. Cover with another slice of bread. Lay a piece of ham on the second slice. Cover with another slice of bread. Lay a slice (or more) of cheese and top with another slice of bread. Secure each corner of the sandwich with a wooden toothpick. Repeat for the remaining ingredients.
Holding a clubhouse sandwich securely in your hands, roll all sides in the beaten eggs. Then dredge all sides in bread crumbs.
With kitchen tongs, lower the sandwich in hot oil. The sandwich will not sink so you will need to turn it over after a few seconds to brown the other side. The frying part should not take more than a few seconds since you don’t want the oil to reach the insides of the sandwich; you just want to brown the outside.
As soon as lightly browned, carefully lift the sandwich from the hot oil with kitchen tongs and drain on lots of paper towels.
Do the same thing with the remaining clubhouse sandwiches.
Using a bread knife, cut each cliff sandwich into four triangles. Remove the toothpicks and serve warm.