Combine the milk, water, and salt in a large, heavy-bottomed pot over medium-high heat and bring to a boil. Once the milk mixture comes to a boil, gradually add the cornmeal while continuously whisking. As soon as all of the cornmeal has been incorporated, decrease the heat to low and cover.
Remove lid and whisk frequently, every 3 to 4 minutes, to prevent grits from sticking or forming lumps; make sure to get into corners of pot when whisking. Cook until mixture is creamy, 20 to 25 minutes.
Remove from heat, add the pepper and butter, and whisk to combine. Gradually whisk in the cheese a little at a time. Serve immediately.