In a stand mixer bowl, combine warm water, yeast, and brown sugar. Let sit for 5-10 minutes until frothy. Add flour, melted butter, and salt. Knead with a dough hook for 5-7 minutes until smooth. Cover and let rise in a warm place for 1 hour, or until doubled.

Pat hot dogs dry. Slice a slit lengthwise down the center of each hot dog. Cut cheese sticks to fit and press a piece of cheese into each slit.

Punch down the risen dough and divide it into 8 equal pieces. Roll each piece into a long rope (about 18-20 inches). Tightly wrap one rope of dough around each cheese-stuffed hot dog, pinching the ends to seal.

Preheat oven to 425°F (220°C). Bring a large pot of water to a boil and stir in the baking soda. Boil each pretzel dog for 30 seconds, then remove with a slotted spoon and place on a parchment-lined baking sheet.

Brush each pretzel dog with egg wash and sprinkle with coarse sea salt. Bake for 12-15 minutes until deep golden brown. Let cool slightly before serving.