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In a mixing bowl, combine the potatoes, parmesan cheese, egg, and ¼ cup of breadcrumbs. Mix thoroughly. The mixture should be somewhat stiff and easy to form into patties. If it is still too “sticky” add some additional breadcrumbs.
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Form the mixture into 6 equal sized patties, about ¾” thick.
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Place the remaining breadcrumbs on a plate or shallow bowl. Coat each side of the mashed potato patties with the breadcrumbs, patting them in slightly.
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In a large nonstick skillet, add enough olive oil to generously coat the bottom. Heat on medium-high until the oil shimmers.
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Add the patties to the hot oil. Be careful -- the oil may spatter! Cook the patties 3-4 minutes per side, until golden brown and heated through.
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Remove the patties from the pan and season with salt and pepper to taste.