Bring the water to a boil in a medium, heavy-bottomed saucepan. Add 1 teaspoon kosher salt if desired.
Slowly whisk in the stone-ground grits in a thin stream, stirring constantly to prevent lumps.
Reduce the heat to low. Simmer gently for 20 to 25 minutes, stirring often, until the grits are tender and creamy. Add a splash of hot water if they become too thick.
Turn off the heat and stir in the butter until melted.
Add the sharp cheddar a handful at a time, stirring until smooth. Taste and add more salt if needed. Serve immediately, topped with an extra pat of butter if desired.