Heat a large saucepan over medium heat. Add the butter, swirling it around to help it melt.
Add the flour to the pot, and whisk it well to combine with the butter.
Cook the mixture over medium heat, whisking constantly until the mixture has started to turn blonde in color.
Add ½ cup of the milk and whisk until it is thoroughly combined with the flour and butter mixture - it will be very thick.
Continue to add the milk, ½ cup at a time, and whisk well after each addition.
Once all of the milk has been added, add the pepper and ground mustard powder.
Bring the sauce up to a low simmer and allow to cook until thickened - 2-3 minutes. Stir often to prevent burning.
Remove the saucepan from the heat and add the shredded cheddar cheese. Stir until it is completely incorporated into the milk, butter, and flour mixture.
Taste and adjust salt and pepper to your taste.
Serve immediately over vegetables like broccoli or cauliflower, or as a base for homemade macaroni and cheese.