Peel and chop the potatoes into chunks, then place them in a large saucepan filled with cold water. Bring to a boil over medium-high heat and cook until tender, about 15-20 minutes. Drain well and set aside.
While the potatoes are cooking, grate or finely chop the carrots and set aside. Sauté the chopped onion in a little oil until golden and fragrant, about 5 minutes, then let it cool slightly.
Once the potatoes are cool enough to handle, mash them thoroughly in a mixing bowl until smooth and creamy. Add the grated carrots, sautéed onion, chopped parsley, and season with a pinch of salt and pepper. Mix everything together until well combined.
Shape the mixture into small logs or patties, about the size of a golf ball or small fist. If the mixture feels too loose, refrigerate for 10 minutes to help it firm up.
Pour the beaten egg into a shallow dish. Place the panko breadcrumbs in another shallow dish. Dip each shaped croquette into the egg, then roll in the breadcrumbs until evenly coated.
Heat the oil in a frying pan over medium heat until bubbling and hot, about 180°C (356°F). Carefully place the coated croquettes into the hot oil, working in batches to avoid overcrowding. Fry until golden brown and crispy, about 3-4 minutes per side.
Use tongs or a slotted spoon to transfer the cooked croquettes onto paper towels to drain excess oil. Let them rest for a minute to crisp up further.
Serve the croquettes hot, garnished with extra parsley or a squeeze of lemon if desired. Enjoy their crunchy exterior and tender, flavorful interior in every bite.
