In a large mixing bowl, dissolve 2 teaspoons of instant rise yeast in 1 ½ cups of lukewarm water along with 2 tablespoons of brown sugar and 1 teaspoon of salt. Mix in 4 cups of all-purpose flour and 4 tablespoons of melted unsalted butter to form a slightly sticky dough.

Dust a clean surface with flour and transfer the dough onto it. Knead for about 2 minutes until smooth and elastic. Cover with a kitchen towel and let rest for 10 minutes.

Divide the dough into 8 equal pieces, roll each into a rope about 1/2 inch thick, cut into 1.5-inch pieces, and arrange on a parchment-lined baking sheet.

In a large pot, bring 4 cups of water to a boil and stir in ¼ cup of baking soda. Boil each pretzel piece for 20 seconds using a slotted spoon to remove them.

Preheat your oven to 425°F. Place boiled pretzel pieces back onto the baking sheet, brush with beaten egg, sprinkle with coarse salt, and bake for 12-15 minutes until golden.

In a saucepan over medium heat, melt 4 tablespoons of unsalted butter, whisk in ¼ cup of all-purpose flour, add 2 cups of warm milk gradually, and then incorporate 2 cups of shredded cheese until melted and smooth.