Use a box grater to grate the potatoes (no need to peel), then place into a large bowl. Fill with enough cold water to comfortably cover the shredded potato, then use your hand to swish everything around. The water should turn cloudy from the excess starch.
A handful at a time, grab the potato, squeeze out excess water, then place in the centre of a large, clean tea towel. Twist it into a ball over the sink and squeeze out as much moisture as possible. For extra insurance, I then like to untwist the ball and press down kitchen roll/paper towels to remove any excess water. If the potato is at all soggy, it won't bind or crisp up on the chicken.
Place the now dry shredded potato in a large dish and combine with the parmesan (do not add salt). Place a dish with the beaten egg to the left of it, then place the flour + the seasonings in a large dish to the left of that.
Carefully slice the chicken breasts right through the centre to create 4 even-sized cutlets. If they're still thick and/or uneven, consider using cling film and a rolling pin to gently pound them to level them out.
One by one, coat the cutlets in the flour, then egg, then the shredded potato. You'll need to press the potato into the chicken to create a flat, even coating of potato that completely covers the chicken.
Add enough oil to comfortably fill the base of a large (preferably heavy-based) pan. Turn the heat to medium-high and heat the oil to 180C/350F (a shred of potato should rapidly sizzle). Carefully add the cutlets and fry both sides for around 3-4 minutes until golden and crispy with the chicken cooked through the centre (safe internal temp of chicken is 74C/165F). Lower the heat/and or add in more oil if they look like they're burning/cooking too quickly (you can peek underneath to check). Place on a wire rack with paper towels underneath to catch excess oil. You may need to work in two batches.
Serve the chicken with a good pinch of salt (remember the potato isn't seasoned), then tuck in and enjoy!