Bring salt and water to a boil in a 2-quart saucepan. Add the grits while whisking, then bring the mixture back to a boil. Reduce heat to low, cover, and continue cooking until you reach the desired consistency: runny with some grit still present, about 30 minutes; thick and creamy but flowing, about 45 minutes; or stiff and able to hold its shape but barely flowing, about 1 hour. Whisk and scrape bottom well every 2 to 3 minutes as you cook. If your grits start to get too thick, add a tablespoon or two at a time of additional water.

Remove from heat and stir in butter until it has completely melted and been absorbed. If necessary, add more salt to the dish.