In the bowl of a stand mixer fitted with a dough hook, or in a large bowl, mix together the flour, yeast, salt, 1 tablespoon of melted butter, brown sugar and 1 cup of very warm water.
Once the ingredients are combined, knead for 5-6 minutes by machine, or 10-12 by hand.
The dough should clear the sides and bottom of the bowl and be very soft and smooth, but not overly sticky. If it’s too sticky add a bit more flour, a tablespoon at a time, if it’s really dry and not tacky at all, add a little more water a tablespoon at a time.
Move the dough to a lightly oiled bowl and let rest for 30-60 minutes, the bowl covered tightly with plastic wrap, until it has doubled in size.
Preheat the oven to 450°F and line a baking sheet with parchment paper.
Combine 10 cups of water with 2/3 cup of baking soda in a large pot, and bring it to a boil.
Meanwhile, punch down the dough, and divide it into 10 equal pieces. Let the pieces rest for 5 minutes, uncovered.
Roll each piece out into a long rope, about 3x the length of the hot dogs.
Wrap each piece around a hot dog, leaving the ends of the hot dog uncovered.
Tuck in the ends of the dough and pinch the seam to seal it and keep it from unraveling.
Boil the wrapped hot dogs, two at a time, for 30 seconds. Place the boiled pretzel dogs on the prepared baking sheet.
Brush the pretzel dogs with melted butter and sprinkle with coarse or kosher salt.
Bake them for 12-13 minutes, until they are a deep golden brown.