Bring the water to a boil in a large saucepan.
Stir in the grits and salt. Reduce the heat to medium low. Boil the grits for 30 minutes, stirring frequently.
Remove the grits from the heat and stir in the cheese and egg.
Pour the grits into a foil lined 9 x 11 x 2-inch baking dish. Allow the grits to cool completely at room temperature. Then cover the dish and refrigerate the grits for 8 hours.
Before cutting the grits, gently lift them out of the pan with the foil lining. Lay them on a flat surface. Cut them into squares or use a round cookie cutter to cut them into disks. Try to keep the grit cakes under 2 ½ inches wide to ensure they hold together during frying.
Sprinkle the flour into a shallow dish. Carefully coat the grit cakes with the flour. Place them on a cookie sheet and return them to the refrigerator for 15 minutes.
In the meantime, heat the oil in a large skillet over medium high heat. Once the oil is hot, return the heat to medium or medium low. Fry the grit cakes in batches, 2 - 2 1/2 minutes per side, until golden brown.
Drain the grit cakes on a layer of paper towels.