Weigh your flour and combine the flour, yeast, sugar and salt in the bowl of a stand mixer. Add the melted butter, then stir in the water to make a soft dough. You can use the paddle attachment or just stir with a heavy-duty scraper or spoon.
Attach dough hook and knead until dough is smooth and pulls away from the side of the bowl, approximately 4 to 5 minutes. Cover the bowl with plastic wrap and let sit in warm spot in the kitchen until doubled (45 minutes to an hour).
Preheat the oven to 450 degrees F. Line a half-sheet pan with parchment paper and lightly brush with the vegetable oil. Set aside.
Meanwhile, turn the dough out onto a pastry mat. If you feel like you need to use a little flour, go ahead. However, if you used bread flour and kneaded the dough long enough you should be able to shape the dough easily without any extra flour or grease. Pat it into a rectangle somewhere around 6x10 inches. Using a knife, slice lengthwise to make 6 10-inch strips.
Move all of the dough strips out of the way and work one strip at a time so you'll have plenty of room for rolling your pretzels. Roll each strip of dough into a rope that's somewhere around 20 to 24 inches. Make a U-shape with the rope, holding the ends of the rope, cross them over each other, make a twist, and press onto the bottom of the U in order to form the shape of a pretzel. Set on prepared pan.
Bring the water to a boil and very gradually add the baking soda. It will fizzle up.
Place the pretzels into the boiling water, one at a time, and boil each for about 20-30 seconds. Remove them from the water using a large flat slotted spatula and letting water drip off. Return to the half sheet pan, brush the top of each pretzel with the beaten egg mixture and sprinkle with your choice of salt. Bake until dark golden brown in color, about 15 minutes (I bake up to 20).
Notes
I've always used 1 teaspoon of Morton kosher in the dough, but I think even less (around ¾ teaspoon) would be okay.