In a small 2- to 3-quart saucepot, add 2 tablespoons of unsalted butter and turn the heat to low-medium.
Once melted, stir in the butter and mix until combined to make a white roux. This only takes 2-3 minutes.
Next, pour in 2 cups of whole milk and, using a whisk, mix until the mixture is smooth and the roux is fully incorporated. Cook while occasionally stirring for 4 to 5 minutes over low-medium heat, until the mixture thickens and coats the back of a spoon.
Remove the pot from the burner and whisk in 8 ounces of shredded sharp cheddar cheese until it is completely incorporated and smooth. Finish with ½ teaspoon of dry mustard, salt, and ground white pepper. Cover with a lid to keep warm.
In the meantime, add the trimmed broccoli florets to a large pot of boiling, salted water and cook for 2 ½ to 3 minutes, or until the florets are tender and the stalk is still gently firm.
Strain immediately and serve in a bowl with the cheese sauce over top.
If you have any leftover cheese sauce, let it cool, then cover and refrigerate for up to 5 days.
