In the bowl of a stand mixer, combine the warm water and sugar. Sprinkle the yeast on top and let it sit for about 5 to 10 minutes until it becomes foamy. This indicates the yeast is alive and ready to work.
Add the melted butter, salt, and flour to the yeast mixture. Using a dough hook, mix on low speed until the ingredients are combined. Increase the speed to medium and knead the dough for about 5 minutes until it is smooth and pulls away from the sides of the bowl.
Place the dough in a lightly oiled bowl and cover it with a clean kitchen towel. Let it rise in a warm, draft-free spot for about 30 to 45 minutes, or until it has doubled in size.
Preheat your oven to 450°F and line two baking sheets with parchment paper. Turn the dough onto a lightly oiled surface and divide it into 12 equal pieces. Roll each piece into a long rope, then twist into a traditional pretzel shape, pressing the ends firmly to secure.
Bring 10 cups of water and the baking soda to a rolling boil in a large pot. Carefully drop each pretzel into the boiling water for 30 seconds. Remove with a slotted spatula and place onto the prepared baking sheets.
Brush the top of each pretzel with the beaten egg wash and sprinkle generously with coarse salt. Bake for 12 to 15 minutes, or until the pretzels are dark golden brown and firm to the touch.
While the pretzels bake, melt 2 tablespoons of butter in a saucepan over medium heat. Whisk in the flour and cook for 1 minute. Slowly whisk in the milk and beer. Once the mixture thickens, reduce heat to low and stir in the shredded cheese, garlic powder, and cayenne until melted and smooth.