In a heavy 2 or 3 quart pot, bring the salted water to a boil over high heat.
Gradually whisk in the cornmeal, stirring continuously to prevent lumps.
Reduce the heat to low and stir in the butter.
Allow the mixture to cook gently, stirring frequently until it begins to thicken. Cover the pot and let it simmer for 25 to 30 minutes, stirring occasionally.
Once the polenta is tender and creamy, stir in the pumpkin puree, Pecorino cheese, and nutmeg.
Adjust salt to taste. Enjoy your creamy pumpkin polenta!