In a large mixing bowl, dissolve instant yeast in 1 cup of lukewarm water, whisking until well combined. Mix in 2 tablespoons of melted unsalted butter, 1 tablespoon of brown sugar, and 1 teaspoon of salt until dissolved. Gradually incorporate 4 cups of all-purpose flour, stirring until a thick dough forms. Knead the dough on a floured surface for about 5 minutes until smooth and elastic.

Cover the dough with a clean kitchen towel and let it rest for 10 minutes at room temperature.

Preheat your oven to 425°F (220°C) and bring 4 cups of water to a rolling boil. Add ¼ cup of baking soda to the boiling water.

Divide the rested dough into six equal pieces and roll each into a rope approximately 24 inches long, twisting it into your desired pretzel shape.

Carefully place the shaped pretzels into the boiling baking soda water for about 20 seconds, then remove them with a slotted spoon.

Brush each pretzel with a beaten egg wash and sprinkle coarse salt on top. Bake in the oven for 12-15 minutes or until golden brown.

In a saucepan over medium heat, warm 1 cup of whole milk until just steaming.

In a separate pan, melt 2 tablespoons of unsalted butter, then whisk in 2 tablespoons of flour to create a roux, cooking for 1 minute.

Gradually whisk in the warm milk until the mixture thickens and then add 2 cups of grated sharp cheddar cheese, stirring until melted and smooth.