Combine the warm water, sugar, and kosher salt in the bowl of a stand mixer and sprinkle the yeast on top. Allow to sit until the mixture begins to foam, about 5 minutes. Add the flour and, using the dough hook attachment, mix on low speed until well combined. Change to medium speed and knead until the dough is smooth and pulls away from the sides of the bowl, 4 to 5 minutes. Remove the dough from the bowl, clean the bowl, and then grease it well with vegetable oil. Return the dough to the bowl, cover with plastic wrap, and sit in a warm place until the dough has doubled in size, 50 to 55 minutes.
Heat oven to 350ºF.
Line 4 half-sheet pans with parchment paper and lightly brush with vegetable oil. Set aside.
Place enough water in a large roasting pan to come 1/3 of the way up the sides of the pan and bring to a boil over high heat.
In the meantime, turn the dough out onto a slightly oiled work surface and divide into 1-ounce portions, approximately 36 pieces. Roll out each piece of dough into a 14- to 15-inch long stick and place on the sheet pans, 1/4 inch apart. Cover the unused dough with a slightly damp towel to help prevent drying out.
Gently place 6 to 8 pretzels at a time into the water for 30 seconds. Remove the pretzels back to the sheet pan, brush with the egg wash, and sprinkle with the pretzel salt. Bake until golden brown in color and hard, 55 to 60 minutes. Transfer to a wire rack to cool for at least 20 minutes before serving. Store in an airtight container for up to 1 week.