Make the soft-boiled eggs: Fill a medium saucepan with water and bring to a boil. Carefully add eggs to boiling water and cook for exactly 6 minutes. Immediately remove eggs and place in an ice bath. Carefully crack and roll each egg on a hard surface and peel off shell. Holding eggs under running water can help to remove shells as well.
Assemble the Scotch eggs: Take about 1/3 cup sausage and form into a flat patty, add slices of American cheese and a soft-boiled egg. Wrap the sausage around the egg, making sure the egg is completely covered. Cover the sausage with potato slices then roll in a light coating of breadcrumbs. Place in refrigerator if meat becomes too warm and hard to work with.
Fry the eggs: Preheat a large saucepan filled with oil to 320 to 325 degrees. Using a slotted metal spoon or spider, carefully lower one egg into the oil. Cook for 4 1/2 minutes. Monitor oil temperature while cooking. Place on a paper towel-lined plate to drain excess oil. Cut in half — the center should be gooey and delicious. Enjoy!