01 - In a large bowl, combine the warm water, active dry yeast, and granulated sugar. Allow the mixture to rest for 5 minutes until it becomes foamy, indicating the yeast is active.
02 - Add the all-purpose flour, salt, and melted unsalted butter to the yeast mixture. Mix thoroughly until a cohesive dough forms. Transfer the dough to a clean surface and knead for 5 to 7 minutes until it is smooth and elastic.
03 - Place the kneaded dough into a lightly greased bowl, cover it with plastic wrap or a clean kitchen towel, and let it rise in a warm place for 45 to 60 minutes, or until its volume has doubled.
04 - Preheat your oven to 450°F (230°C). Line two baking sheets with parchment paper to prevent sticking.
05 - In a large pot, bring 10 cups of water and 0.67 cup of baking soda to a rolling boil. This alkaline solution is crucial for the pretzels' characteristic crust and flavor.
06 - Gently punch down the risen dough to release air. Divide the dough into 8 equal pieces. Roll each piece into a 20-inch-long rope, then carefully twist and shape it into a traditional pretzel form.
07 - Carefully immerse each shaped pretzel into the boiling baking soda solution for exactly 30 seconds. Using a slotted spatula, remove the pretzels from the water, allowing excess liquid to drain, and place them onto the prepared baking sheets.
08 - In a small bowl, whisk together 1 large egg with 1 tablespoon of water to create an egg wash. Brush this mixture generously over each pretzel and then sprinkle with coarse sea salt.
09 - Bake the pretzels in the preheated oven for 12 to 14 minutes, or until they achieve a deep golden brown color. Once baked, transfer them to a wire rack to cool slightly.
10 - For the cheese sauce, melt 2 tablespoons of unsalted butter in a medium saucepan over medium heat. Whisk in 2 tablespoons of all-purpose flour and cook for 1 minute, creating a roux.
11 - Gradually whisk in 1 cup of whole milk into the roux. Continue to cook, stirring constantly, until the sauce thickens, which should take approximately 2 to 3 minutes.
12 - Remove the saucepan from the heat. Stir in 1 cup of shredded sharp cheddar cheese, 0.5 cup of shredded mozzarella cheese, 0.5 teaspoon of Dijon mustard, 0.25 teaspoon of garlic powder, and season with salt and black pepper to taste. Stir until the cheeses are completely melted and the sauce is smooth.
13 - Serve the warm, freshly baked pretzels immediately with the creamy cheese sauce for dipping.