Preheat your oven to 400˚F. Line a baking tray with a silicone mat or parchment paper.
In a medium size bowl, mix together the flour, cayenne and salt. Stir in the grated cheese and set aside.
In another medium size bowl, free of any grease, beat the egg whites using an electric hand mixer or a whisk until you have stiff peaks. Fold a quarter of the cheddar mixture into the egg whites. Then add the rest and continue to fold until evenly combined. Using your hands, shape the mixture into small balls, approximately 1 to 1 1/2 inches in diameter. Roll the balls in the breadcrumbs until coated all over. Place on your prepared baking tray. Continue until you have used all of the mixture. You should be able to make about 20 balls that fit on one tray.
Bake the cheese balls in the oven for 10 minutes or until they have puffed up and are turning slightly golden in color. Remove from the oven and serve immediately.
Most hard cheeses will work with this recipe. One exception being hard cheeses that are quite salty, such as mimolette. Being in France I tried this recipe once with this cheese and the end result was very salty cheese balls. Best to stick with hard cheeses such as cheddar, comté, or gouda.
If you think your cheese might be salty, you can take out the salt called for in the recipe.