Preheat oven to 200 degrees Fahrenheit (93 degrees Celsius).
In the bowl of a stand mixer fitted with the paddle attachment, mix the flour, sugar, yeast, and salt on low for 30 seconds. Add the melted butter and water, and mix just until a soft dough forms. Switch to the dough hook and knead on low for 5 minutes.
Transfer the dough to a lightly greased bowl and cover with a clean towel or plastic wrap. TURN THE OVEN OFF and place in the oven to rise for 1 hour.
Fill a large stockpot about 2/3 full of water and place it on the stovetop. Add the baking soda and bring to a boil. While the water is coming to a boil, divide the dough into 16 equal pieces. Roll it out with a rolling pin or use your hands to create a rectangle. Wrap around each half and pinch at the bottom. Make sure that it's sealed together well so that it doesn't come undone while it boils or bakes.
Turn the oven on to 425 degrees Fahrenheit (218 degrees Celsius).
Put the dogs into the boiling water (4 at a time) for 30 seconds, then transfer them onto a baking sheet lined with baking mats or buttered parchment.
Beat the egg and the water in a small mixing bowl and brush over the tops of the dough. Sprinkle with salt (slit the tops if desired, just take care not to go all the way down to the hot dog) and place into the oven. Bake for 12-15 minutes or until deep golden brown. Remove and enjoy warm.
Notes
If you want to stuff the dogs with pickles or cheese, slice them in half lengthwise (taking care not to cut them in half) and place a slice of pickle or cheese inside. Fully wrap the hot dogs with the dough, and follow the directions as written.
To STORE leftover Mini Pretzel Dogs, place them in an airtight container in the fridge, lining the bottom with paper towels to avoid sogginess. Microwave until warmed through.If you have made too many, you can easily FREEZE them. Assemble and bake as normal, then let them cool completely. Layer in a freezer-safe bag and freeze for up to a month.REHEAT on a tray in the oven at 325 degrees Fahrenheit (163 degrees Celsius) for 12-14 minutes or until heated through.