Bring the water to a boil over high heat in 4 quart pot. When boiling, add basil, salt, and pepper.
Whisk in plent in a steady stream, stirring with a wooden spoon.
It will start to thicken immediately. Reduce the heat to low, add the goat cheese, and cook, stirring for another 6 to 8 minutes. The polenta will be firm.
Pour a 3/4 inch thick layer of polenta onto a large sheet pan with at least a 1 inch rim. Make sure the pan is covered evenly.
Let cool, then cut into shapes with 3 inch heart-shaped cutter.