In a large mixing bowl, combine activated instant rise yeast, melted unsalted butter, brown sugar, and salt. Gradually mix in all-purpose flour until a sticky dough forms. Let the dough rest for about 10 minutes.

Turn the dough out onto a floured surface. Knead it for about 2 minutes, forming a smooth ball. Cover with a towel and let it relax for another 10 minutes.

Preheat your oven to 425°F (220°C). In a large pot, bring about 10 cups of water mixed with 2/3 cup of baking soda to a rolling boil. Cut dough into sections, roll each into a long rope, twist into pretzel shapes, and boil for 20 seconds.

Brush the boiled pretzels with whisked egg, sprinkle with coarse salt, and bake for 12-15 minutes until golden brown.

While the pretzels bake, melt 2 tablespoons of butter in a saucepan over medium heat. Whisk in flour and cook for 1-2 minutes. Gradually whisk in warm milk until well combined, then stir in cheddar cheese until melted.

Let the pretzels cool slightly before serving with the cheese dip.