In a medium saucepan, bring the water and salt to a boil. Add the polenta and whisk continuously until thickened (like oatmeal).
Reduce the heat to low and simmer. Let the polenta cook, uncovered, for 50 minutes, stirring every 10 minutes. If the polenta gets too thick, add a splash of milk, and stir.
Once the polenta is tender, add the butter, remaining milk, and ½ cup cheese. Add more salt, if needed.
Serve the polenta and top with remaining Parmesan cheese.