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Bring 4 cups of water to a boil in a heavy 4-quart saucepan. Add the salt, then slowly add the grits in a thin, steady stream, stirring constantly with a wooden spoon. Reduce the heat to low and simmer, stirring occasionally, until the grits thicken, about 5 to 7 minutes.
Add the half-and-half and butter to the grits and stir. The mixture will seem thin but it will thicken as it cooks. Bring to a simmer, stirring occasionally. Cover the pot, reduce the heat to low, and cook, stirring occasionally for 45 minutes, until very smooth and creamy. Off the heat, stir in the Cheddar, scallions, and pepper. Season to taste and serve hot with a sprinkle of grated cheese and scallions.