Bring 4 cups of water to a boil in a medium saucepan. Add the salt, then slowly add the grits in a thin steady stream. Stir constantly with a wooden spoon. Reduce heat to low and simmer, stirring occasionally, until grits thicken, 5-7 minutes.
Add the half and half and butter to the grits and stir. 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 cheese, green onions, and pepper. Season to taste and serve hot.