Melt the butter in a saucepan over medium heat. (Watch carefully. Once melted, the butter will brown quickly. Lower the heat to medium-low if starting to brown or if making a half batch.)
Add the flour or cornstarch and whisk or stir for about 1 minute over medium heat—the mixture should be simmering. Slowly whisk in the milk, and then add the salt and pepper.
Cook, stirring regularly, until slightly thickened, about 4 minutes. Remove from the heat, add the cheese, and stir until melted and smooth. Serve immediately.