Heat salted water to a boil over high heat in a heavy-bottomed 2 or 3 quart pot.
Slowly add the polenta into the boiling water, whisking while you add it to prevent any lumps from forming.
Lower the heat to a very low simmer and add the butter.
Continue to whisk the polenta until it starts to thicken. Then cover the pan and let the polenta cook for 25 to 30 minutes, stirring occasionally with a wooden spoon so the polenta doesn't stick to the bottom of the pot.
The polenta should be soft and tender when done. If not, let it cook a few minutes longer.
Add the pumpkin, Pecorino cheese and nutmeg, stir it into the polenta.