In a medium saucepan over medium heat, melt 2 tablespoons unsalted butter.
Add 2 tablespoons all-purpose flour and whisk continuously until smooth and the raw flour taste cooks off, about 1 minute.
Gradually pour in 1 cup whole milk in a slow stream while whisking constantly to prevent lumps; continue whisking until fully combined.
Cook the mixture over medium heat, whisking frequently, until it thickens and coats the back of a spoon, about 5 minutes.
Reduce the heat to low. Add 1 1/2 cups shredded sharp cheddar cheese in small handfuls, stirring until each addition is fully melted and the sauce is smooth.
Stir in 1/4 teaspoon salt and 1/4 teaspoon black pepper, remove from the heat, and serve immediately.