Boil the Potatoes: In a large pot, bring water to a boil. Add the cubed potatoes and a pinch of salt. Cook until they are fork-tender, about 15-20 minutes. Drain well.
Mash the Potatoes: Transfer the drained potatoes to a large bowl. Mash them until smooth using a potato masher or a fork, letting any steam escape.
Mix Ingredients: To the mashed potatoes, add the shredded chicken, steamed broccoli, breadcrumbs, grated cheese (if using), egg, garlic powder, onion powder, salt, and pepper. Mix everything until well combined.
Shape the Cakes: Using your hands, form the mixture into small patties, about 2-3 inches in diameter. Place them on a lined baking sheet or plate.
Heat the Oil: In a large skillet, pour enough vegetable oil to coat the bottom of the pan and heat over medium heat.
Fry the Potato Cakes: Once the oil is hot, carefully add the potato cakes to the skillet. Cook for about 4-5 minutes on each side, or until they are golden brown and crispy. You may need to work in batches depending on the size of your skillet.
Drain Excess Oil: Once cooked, transfer the cakes to a plate lined with paper towels to absorb any excess oil.
Serve Hot: These cakes are best served hot, accompanied by your favorite dipping sauce or a fresh green salad.