Peel and chop the potatoes. Boil them in salted water until soft and fork-tender.
Drain the potatoes well and mash them in a large bowl while still warm.
Add grated carrot, butter, chopped parsley, and salt to the mashed potatoes. Mix thoroughly.
Stir in the grated cheese and mix until the cheese is evenly incorporated.
Let the mixture cool slightly. Then shape it into sticks (about finger-sized).
Heat a skillet over medium heat and lightly grease it with oil or butter.
Pan-fry the sticks until golden and crisp on all sides. Alternatively, bake them at 400°F (200°C) for 20–25 minutes until golden.
Serve warm with your favorite dipping sauce.