In a large bowl, combine warm water, yeast, and sugar. Let sit for about 5 minutes until frothy.

Add flour and salt to the yeast mixture. Stir until a dough forms, then knead on a floured surface for 5-7 minutes until smooth and elastic.

Place the dough in a greased bowl, cover with a kitchen towel, and let rise in a warm place for about 1 hour until doubled in size.

Preheat the oven to 450°F (230°C) and line a baking sheet with parchment paper.

In a large pot, bring 10 cups of water to a boil and add baking soda.

Punch down the risen dough and divide it into 8 equal pieces. Roll each piece into a 24-inch long rope and shape into pretzels.

Boil each pretzel in the baking soda water for about 30 seconds, then remove with a slotted spoon and place on the prepared baking sheet.

Brush the pretzels with the beaten egg and sprinkle with coarse sea salt.

Bake the pretzels for 12-15 minutes until golden brown.

While the pretzels are baking, melt butter in a saucepan over medium heat. Stir in flour and cook for 1 minute to create a roux.

Gradually whisk in milk, stirring constantly until thickened (about 3-5 minutes). Remove from heat and stir in cheese and Dijon mustard until smooth. Season with salt and pepper to taste.

Serve the warm pretzels with the cheese dip on the side.