Combine the water, sugar and kosher salt in the bowl of a stand mixer and sprinkle the yeast on top. Allow to sit for 5 minutes or until the mixture begins to foam. Add the flour, butter, Italian seasoning and parmesan and, using the dough hook attachment, mix on low speed until well combined. Increase the speed to medium and knead until the dough is smooth and pulls away from the side of the bowl, approximately 4 to 5 minutes.
Remove the dough from the bowl, clean the bowl and then oil it well with vegetable oil. Return the dough to the bowl, cover with plastic wrap and sit in a warm place for approximately 50 to 55 minutes or until the dough has doubled in size.
Preheat the oven to 450 degrees F. Line 2 half-sheet pans with parchment paper and lightly brush with the vegetable oil. Set aside. Meanwhile, bring the 10 cups of water and the baking soda to a rolling boil in an 8-quart saucepan.
In the meantime, turn the dough out onto a slightly oiled work surface and divide into 10 equal pieces. Roll into ropes, and then roll the rope around an entire hot dog, making sure to secure the ends.
Place onto 2 sil-pat lined sheet pans.
Place the pretzeldogs into the boiling water, 1 by 1, for 30 seconds. Remove them from the water using a large flat spatula. Return to the half sheet pan, brush the top of each pretzel with the beaten egg yolk and water mixture and sprinkle with the pretzel salt. Bake until dark golden brown in color, approximately 12 to 14 minutes.
Transfer to a cooling rack for at least 5 minutes before serving