Preparation

Soak the polenta

• Put the polenta in a 4-quart heavy-duty saucepan and whisk in the boiling water. Cover and let sit at room temperature for at least 8 and up to 12 hours.

Cook the polenta

• In a 2-quart saucepan, bring the broth and milk to a boil. Whisk the broth mixture and 1 tsp. salt into the polenta, loosening it and breaking up any clumps. Bring to a boil over medium-high heat, whisking once or twice, about 5 minutes.
• Turn the heat down to maintain a simmer and continue cooking, whisking constantly, until the polenta thickens from soupy to porridge-like, about 2 minutes.
• Turn the heat down to low. Cover and cook, stirring vigorously and scraping the bottom of the pan with a wooden spoon or spatula every couple of minutes. When you can see the bottom of the pan as you drag the spoon across it, 5 to 10 minutes later, begin tasting the polenta; it’s done when it’s thick, creamy, and tender. It should be granular, but not gritty.
• Remove from the heat, and stir in the cheeses, butter, and 1/2 tsp. pepper until the butter is melted. Season to taste and serve right away, passing more Parmigiano at the table.

Make Ahead Tips

After soaking the polenta at room temperature, you can refrigerate it, covered, for up to 2 days.