In a shallow plate or pie dish, mix the breadcrumbs with salt, pepper, and cayenne pepper. In a large bowl, stir together the potatoes and cheese. Scoop out 1/4 cup of the potato mixture and form it into a small patty. Gently press the patties into the seasoned breadcrumbs, coating both sides, and set aside.
In a large skillet over medium heat, add the oil. When it is hot, add the potato cakes in 1 layer and cook until golden brown and crispy, about 2 minutes per side. Drain on brown paper and sprinkle with more salt and pepper while hot.