Combine the water, sugar and yeast in the bowl of a stand mixer. Let sit for 5 minutes or until the mixture begins to foam.
Add the flour, salt, and butter. Combine with a spoon or the paddle until it start to come together, then switch to the dough hook. Mix with the dough hook on low speed until well combined.
Change to medium speed and knead until the dough is smooth and pulls away from the side of the bowl, approximately 4 to 5 minutes. You may need to add a bit more flour, but be aware that this is a pretty soft dough. You should be able to touch it without it being impossible to get off your fingers, though.
Transfer the dough to a lightly oiled bowl. Cover and let rise in a warm place for about 1 hour, or until the dough has doubled in size. I like to put mine in an oven that has been turned to 170? (warm) for about a minute, then turned off.
Preheat the oven to 450. Line 2 baking sheets with parchment paper and lightly brush with the vegetable oil. Set aside.
In a large pot, bring the 10 cups of water and the baking soda to a rolling boil. Meanwhile, knead down the dough, turn it out onto a slightly floured work surface, and divide into 8 equal pieces.
Roll out each piece of dough into a rope that's about 20 to 24-inches long, depending on how big your hot dogs are. Starting at one end, wrap the dough around the hot dog, pinching each end together so that it's sealed. Place onto the pan and repeat with the remaining dough and hot dogs.
Place the pretzel dogs into the boiling water two at a time and boil for 30 seconds. Remove them from the water using a large flat slotted spatula. Return the boiled pretzel dogs to the pan, brush the tops with melted butter and sprinkle with the sea salt. (I forgot to do this butter and salt step until after I had baked them. It worked fine.)
Bake until dark golden brown in color, approximately 12 to 14 minutes. Transfer to a cooling rack for at least 5 minutes before serving.
If you have extra dough, you can shape them into dinner rolls, cut a slit in the top, let rise for another 20-30 minutes, boil, and bake at 450 until golden.