Combine the warm water and sugar in the bowl of a stand mixer and sprinkle the yeast on top. Let sit for 5 minutes or until the mixture begins to foam.
Add the flour, butter, and salt and mix on low speed using the dough hook until well combined. Increase to medium speed and knead until the dough is smooth and pulls away from the side of the bowl, about 4 to 5 minutes. You may need to add an additional bit of flour to help the dough come together to clear the bowl. Add additional flour a spoonful at a time until the dough clears the edge of the bowl and then continue kneading.
Remove the dough from the bowl and place in a clean, oiled bowl. Cover with plastic wrap and sit in a warm place for approximately 50 minutes or until the dough has doubled in size.
Bring the 10 cups of water and the baking soda to a rolling boil in a deep pot.
In the meantime, separate the dough into 10 equal pieces. Use your hands to roll and stretch the pieces of dough into ropes, about 12 inches long.
Wrap one piece of dough around each hot dog, leaving just a bit of the dog peeking out at the ends. Press the ends of the dough together to hold in place.
Preheat the air fryer to 370 degrees for 5 minutes while you boil the pretzel dogs.
Place the pretzel dogs, one at a time, into the boiling water for 30 seconds. Only boil as many pretzel dogs as will fit in your air fryer basket at one time.
Remove them from the water using a slotted spoon. Place on a parchment lined baking sheet and brush the top of each pretzel with the beaten egg yolk and water mixture and sprinkle with the pretzel salt.
Spritz the basket with oil spray and arrange the pretzel dogs in the basket, leaving space between each for air to flow.
Air fry until dark brown in color, approximately 6-7 minutes. Transfer to a cooling rack for at least 5 minutes before serving.
Repeat the boiling and air frying process with the remaining pretzel dogs while the first batch is cooling.
Serve hot.
Notes
You may need to add a bit more flour than called for, depending on humidity levels. The dough should clear the side of the mixing bowl and not be overly sticky to the touch - just a bit of residue left behind if you poke the dough. Add additional flour a spoonful at a time and continue kneading until the dough is workable.If you don’t have a stand mixer, you may try kneading this dough by hand.Boiling the pretzel dogs in the baking soda water is what gives pretzels that distinctive pretzel flavor along with the crispy exterior and chewy interior. We do not recommend skipping this step.If you'd prefer to bake these in the oven so you can make the whole batch at once, bake at 450 degrees on a sheet pan for 10-12 minutes.