Line a quarter sheet pan or 9x13 pan with parchment. Grease parchment thoroughly.
In a medium saucepan, whisk together cornmeal, water, half and half (or milk), and salt. Turn heat to medium.
Whisk constantly until mixture begins to simmer. Turn heat to low. Stir intermittently until mixture thickens significantly and begins to hold a bit of shape/ pulls away from the edges of the pan. This takes about 15-20 minutes.
Stir in 1/2 cup cheese.
Pour polenta out into prepared pan. Spread to distribute evenly. Let rest on a wire rack until cooled and set. (At least 20 minutes)
When polenta is cooled, cut into squares.
Preheat gas grill to 450-500 degrees Fahrenheit. Or preheat cast iron grill pan to high heat.
Brush one side of the polenta squares with olive oil. Arrange squares on grill, placing on a diagonal to create off-set grill marks. Press polenta into the grill with a spatula, using light pressure. Brush remaining side of polenta with additional olive oil.
Cover (gas grill only, unncessary if using grill pan) and grill for about 5 minutes, or until polenta easily lifts from grill grates with a spatula. Flip and grill for about another 5 minutes.
Arrange on platter and sprinkle with remaining cheese, and minced herbs.