White grits tend to have a fine consistency and hence a relatively short cooking time compared with some other varieties. They are not the same as instant grits.
Bring water to a boil in a medium pot. Whisking constantly, slowly add grits to water and cook until mixture starts to thicken, about 1 minute. Reduce heat to medium-low, and simmer gently, stirring occasionally, until grits are smooth, thick, and fall easily from the spoon, about 25 minutes.
Add cheese, salt, and cayenne, and stir just until cheese melts. Serve immediately, or cover and keep grits warm over low heat, stirring occasionally, for up to 1 hour (you may need to adjust the consistency with additional water just before serving).