Bring the water and salt to a boil in a large heavy saucepan. Add the grits and continue to simmer over medium heat, stirring constantly, until the grits are cooked and thick like mush, about 15 to 20 minutes. If necessary, add a little more boiling water. Add the rosemary and butter.
Pour the hot grits onto a large plate to make a layer about 1/2-inch deep. Cover and let stand to cool, then refrigerate to chill thoroughly.
When the grits are cold and firm, cut them into rectangular pieces or use biscuit cutters and cut the grits into rounds. Dredge the grit cakes in flour, shaking off any excess flour.
Heat about 1/2 inch of vegetable oil in a large, heavy skillet over medium-high heat. Fry the grits cakes until golden brown on both sides, about 4 to 5 minutes total. Drain on paper towels and sprinkle with salt. Serve hot.