Dredge the patties in flour, shaking off the excess, then dip in the beaten egg, letting the excess drip back into the bowl. Press the patties into the bread crumbs to coat both sides. When the oil is hot, carefully slide the patties into the oil. Fry, turning once, until golden brown and just cooked through, about 8 to 10 minutes. (You will know the patties are cooked through when you press on one and no more juice comes out.) Drain the cooked patties on paper towels.