Shred 8 ounces of sharp cheddar from the block on a box grater into a bowl until you have about 2 cups, lightly packed. Set it next to the stove and measure the milk, salt, pepper, and optional spices ahead of time.
Place a medium saucepan over medium heat and add the butter. Let it melt completely, swirling the pan, about 1 minute, without letting it brown.
Sprinkle the flour over the melted butter, adding the optional dry mustard and cayenne if using. Whisk constantly for 1 to 2 minutes until thick, pale, and slightly bubbly.
Pour in about ¼ cup of the milk and whisk hard until smooth and thick, then gradually add the rest of the milk a splash at a time, whisking constantly and scraping the corners of the pan.
Keep the heat at medium and whisk for 3 to 5 minutes until small bubbles appear at the edges and the sauce coats the back of a spoon. Never let it boil hard.
Slide the saucepan off the burner or turn the burner off completely. Wait about 30 seconds, stirring, so the temperature drops and the sauce looks like a thick, smooth white sauce.
Add the shredded cheddar in three or four handfuls, stirring until each is fully melted before adding the next. If needed, return to very low heat for 15 to 20 seconds, stirring constantly.
Stir in the salt and black pepper, taste and adjust seasoning, and thin with a tablespoon of milk at a time if desired. Serve right away while warm and fluid.