Bring a large pan of lightly salted water to the boil and cook the carrots and potatoes for about 12 minutes until tender. Drain well and mash together until crushed but not completely smooth. Set aside to cool, uncovered, for at least 10 minutes.
While the potatoes and carrots are cooling, add the beaten egg, feta, cumin, parsley, onion and a pinch each of salt and pepper to the pan and mix well to combine. Use flour-dusted hands to form the mixture into 4 patties.
Place the oil in a large non-stick frying pan and shallow-fry the patties gently for about 3 minutes on each side until crisp and golden. Drain on kitchen paper and serve with fried or poached eggs, if desired.