Mix whole grain mustard, Greek yogurt, garlic powder, and cayenne pepper in a small bowl until well blended.
Place 2 beaten eggs into a shallow dish; place potato flakes in a separate shallow dish.
Divide corned beef hash into 4 portions. Form corned beef hash around each egg until the egg is completely encased.
Roll encased eggs into the beaten egg and coat with mashed potato flakes until covered.
Heat oil in a deep-fryer or large saucepan to 375 degrees F (190 degrees C).
Drop 2 eggs into the hot oil and fry until brown, 3 to 5 minutes. Remove with a slotted spoon and place on a paper towel-lined plate. Repeat with remaining 2 eggs.
