In a heavy-bottomed saucepan, bring 6 cups water and 1/2 cup whole milk to a gentle simmer over medium heat. Add 1 1/2 teaspoons salt and 1/4 teaspoon black pepper. Keep the liquid at a bare simmer; do not boil vigorously.

Whisk in cornmeal

Gradually whisk in 1 1/2 cups coarse cornmeal in a steady stream to the simmering liquid to prevent lumps. Continue whisking until all the cornmeal is incorporated and mixture thickens slightly.

Cook polenta, stirring

Reduce heat to low. Cook the polenta at a gentle simmer, stirring frequently with a wooden spoon or spatula to prevent sticking and to break up any lumps. For coarse polenta, cook about 30–40 minutes until grains are tender and the texture is creamy and thick. If mixture becomes too thick while cooking, add hot water (a few tablespoons at a time) to adjust consistency.

Finish with butter and cheese

Once the cornmeal is tender and polenta is creamy, remove from heat. Stir in 2 tablespoons unsalted butter and 1/2 cup grated Parmesan until melted and fully incorporated. Taste and adjust seasoning with additional salt and pepper if needed.

Optional finish / serving

Serve soft and creamy

Spoon the finished polenta into bowls, top with 1 tablespoon extra butter, an optional 1/4 cup extra grated cheese, a drizzle of olive oil (about 1 tablespoon) and sprinkle with chopped fresh parsley or chives. Serve immediately while hot and creamy.

Or cool and slice (set polenta)

To make firm polenta for grilling or frying, pour the hot polenta into a lightly oiled shallow baking dish or rimmed tray, smoothing the top. Let cool to room temperature, then refrigerate until fully set, about 2 hours. Once set, turn out onto a cutting board and slice into squares or wedges. Brush slices with olive oil and grill or fry 2–3 minutes per side until golden and crisp. Serve with extra grated cheese and herbs.