Grab a large bowl and stir together the mashed potatoes, cheese, onion, carrots, egg, salt and flour until combined (start with 2-3 tablespoons and gradually add as needed).
Using a large spoon, divide the mixture into portions, depending on how big you want your pancakes. Roll each portion into a compact ball then flatten it into a pancake (your choice of thickness but I'd recommend a 1/2"). Place some bread crumbs in a shallow dish and dredge each pancake.
Heat some vegetable oil in a large sauté pan over medium heat (thoroughly coat the bottom of the pan). Fry the pancakes, in batches, until they're golden brown and crispy on both sides, 2-3 minutes. Add more oil to the pan as needed between batches (don't overcrowd and don't flip the pancakes too soon or they won't develop a crisp crust).Transfer the pancakes to a paper towel. Serve the pancakes topped with sour cream or dipped in Tzatziki sauce or simple as a side dish, alongside meat and salad.