Bring 6 cups water and the salt to a boil over high heat in a medium (6-quart) heavy-bottom pot. Bring remaining 4 cups water to a simmer over medium heat in a small saucepan.
Gradually add polenta to the large pot while whisking constantly. Switch to a wooden spoon once mixed in with no lumps. Cook until cornmeal has absorbed all of the water, about 3 minutes. Lower heat until only one or two large bubbles break the surface at a time (over medium-low or low, adjusting heat as necessary).
Stir 2 ladles of simmering water into polenta and cook, stirring frequently with a wooden spoon, until water has been absorbed, about 5 minutes. Continue to add 2 ladles of water every 5 minutes, stirring frequently to prevent polenta from scorching and waiting for it to be absorbed before adding more, until polenta is creamy and just pulls away from sides of pot, about 45 minutes. (It may be necessary to adjust heat.)
Stir in butter and season with pepper. Use a damp spoon or ladle to transfer polenta to wide, shallow bowls, and shave cheese over each serving.