In a mixing bowl, combine lukewarm water with instant rise yeast and let it sit for about 5 minutes until frothy.

Stir in melted unsalted butter, brown sugar, and salt until well combined.

Gradually add all-purpose flour, mixing until a dough begins to form.

Transfer the dough onto a floured surface and knead for approximately 2 minutes until smooth and elastic.

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

Preheat oven to 425°F and line two baking sheets with parchment paper.

Divide the rested dough into 8 equal pieces, roll each piece into a 24-inch rope, and shape into pretzels.

Boil 6 cups of water and add in 1/2 cup baking soda.

Carefully drop the pretzels into the boiling solution one at a time, boiling for about 20 seconds.

Remove each pretzel with a slotted spoon and place on the baking sheets.

Brush each pretzel with the beaten egg and sprinkle with coarse salt.

Bake for 12-15 minutes until golden brown and puffed.

In a saucepan, heat whole milk over medium heat, melt unsalted butter, and whisk in flour until golden.

Gradually pour in warmed milk, stirring until thickened, then add shredded sharp cheddar cheese until smooth.