Add the butter to a large pot on the stove top over medium high heat.
When the butter begins to melt, add the milk, half and half, cornmeal, and salt to the pot.
Stir continuously, 10-15 minutes, until the polenta becomes thick and creamy.
Add the cheeses and continue stirring until the cheese has melted into the polenta.
Remove from the heat and serve immediately or add to your favorite dish.
Notes
As the polenta cools, it will firm up. If you want a soft and smooth consistency it should be served immediately. You can reheat solidified polenta back to its creamy state by returning to a pot on the stove and stirring in more milk, until you reach your desired smoothness.