Bring water to a boil in a large pot. Salt it well, then add the polenta gradually, whisking constantly. When the mixture thickens, switch to a wooden spoon and adjust the heat to maintain a bare simmer. Cook, stirring often, until thick, smooth, and creamy, about 40 minutes. Season to taste with salt.

Stir in the Parmesan and the 1 tablespoon Basil Oil. Remove from the heat and stir in the butter.

Spread polenta onto a board or a piece of wax paper; divide into 4 equal portions, and allow diners to garnish as desired with Basil Oil, marinara, pesto, and additional cheese. 

Herb Infused Oils: because the flavor intensity of various herbs varies depending on the time of year, Chiarello recommends making a concentrate, then diluting as necessary depending on use. For soft herb such as basil, parsley, cilantro, or tarragon, use 4 cups firmly packed leaves to 2 cups mild-tasting olive oil. For woody herbs such as rosemary, sage, thyme, oregano, or lemon verbena, use 1 cup firmly packed leaves to 2 cups olive oil.

Puree the herbs and oil in a blender (food processor won't do it) until completely smooth. Transfer to a sauce pan and bring to a simmer over moderate heat; simmer for 45 seconds, then pour through a fine mesh sieve into a bowl. Don't press on the mixture. Immediately strain again through a flat-bottomes paper filter; if it clogs, you may need to change the filter partway through.

Let the filtered oil settle for a few hours, then pour off the clear oil, leaving any dark liquid behind. Store filtered oil in an airtight container; it will stay lively for about a month.