Add the 1 1/2 cups of warmed water, sugar, and salt into the bowl of a standing mixer and stir well. Sprinkle the active dry yeast on top and let this rest for 10 minutes. The mixture should then be super frothy; if not, this means your yeast is spoiled or your water wasn't at the correct temperature, so please start over.
Now, add the flour and melted butter into the bowl. Using a dough hook attachment, mix on low speed until well combined. Then, change the speed to medium and knead the dough for 4 minutes.
In the meantime, grease a large mixing bowl with some vegan butter. After the dough has finished kneading, roll it around in the greased bowl to lightly coat it, then place a piece of plastic wrap directly on top of the dough. Cover the bowl in plastic wrap as well. Let the dough rest in a dark, warm place for 1 hour or until doubled in size.
Preheat the oven to 450 degrees Fahrenheit. Line two large sheet pans with parchment paper and lightly brush them with melted vegan butter. Set aside.
Bring the 10 cups of water and the baking soda to a rolling boil in a medium-large stockpot.
In the meantime, lightly grease your counter with vegan butter. Then, punch the dough once down the center and turn it out onto the greased counter. Cut the dough into 8 equal pieces and place a piece of plastic wrap on top of them. Roll out each piece of dough into a 24-inch rope. Make a u-shape with the rope, then take one end of the rope, bring it down to the bottom opposite side, and seal it into the dough. Next, take the other end of the rope, cross it over to the other bottom side, and seal it into the dough, giving it a traditional pretzel shape. Gently place this on the prepared sheet pan. Repeat this step with the other 7 ropes of dough.
Gently lift a pretzel with a large spatula while also holding it with your other hand, and carefully place it into the boiling water to cook for 30 seconds. Then, gently lift it out of the water using the spatula or even two spatulas to make it easier, and put the pretzel back onto the prepared sheet pan. Repeat this step with all of the pretzels.
Stir the liquid Just Egg with the 1 tbsp of water until combined. Using a pastry brush, brush the entire top and sides of a pretzel with the egg mixture and then immediately sprinkle it generously with coarse sea salt. Repeat with all pretzels.
Bake one sheet pan at a time on the middle oven rack for 10-12 minutes or until medium-dark golden brown, rotating the pan halfway through. Then, transfer them to a wire rack to cool down for at least 5 minutes. Repeat this step with the other sheet pan of pretzels.
When you're ready to eat one, brush it with melted butter if desired, and dip it in vegan nacho cheese or queso, mustard, or whatever your heart desires!