Mix warm water, yeast, and brown sugar in a bowl. Let sit for 5-10 minutes until frothy.

Add melted butter, salt, and 3 cups of flour. Mix until combined. Gradually add remaining flour until a dough forms.

Knead dough for 5-8 minutes until smooth and elastic. Place in an oiled bowl, cover, and let rise for 1 hour until doubled in size.

Preheat oven to 400°F (200°C). Line two baking sheets with parchment and lightly spray with oil. Bring 8 cups of water and baking soda to a rolling boil in a wide pot.

Punch down dough and divide into 8-10 equal pieces. Roll each into a 20-inch rope. Wrap a rope tightly around each dried hot dog, pinching the ends to seal.

Drop 2-3 pretzel dogs at a time into the boiling water for exactly 30 seconds. Remove with a slotted spoon and place on prepared baking sheets.

Brush each pretzel dog with beaten egg and sprinkle with coarse salt.

Bake for 12-15 minutes until deep golden brown. Serve warm with cheese sauce or mustard.