Activate the yeast. Combine the yeast, 1 teaspoon of the sugar, and the warm water in a small bowl. Let rest for 10 minutes to activate the yeast. It should look foamy.
Make the dough. In a stand mixer fitted with the dough hook, whisk together the flour, the remaining ½ cup of sugar, and the kosher salt. Add the oil and the yeast mixture. Mix on medium-low speed for about 5 minutes until the dough is smooth and pulls away from the sides of the bowl.
Let the dough rise. Transfer the dough to a greased bowl and turn to coat all sides. Cover and let rest in a warm place until doubled in size, about 1 hour.
Prepare for shaping. Near the end of the rising time, preheat the oven to 425°F. Line two baking sheets with parchment paper. Bring the 9 cups of water and baking soda to a boil on the stovetop, stirring to dissolve the baking soda.
Shape the pretzels. Punch down the dough to release the air. Turn onto a lightly floured surface and divide into 12 equal pieces. Roll each piece into a rope about 22-24 inches long. Twist each rope into a traditional pretzel shape by lifting both ends, twisting once, then folding the twisted section down and pressing the ends onto the bottom curve.
Boil in the baking soda bath. Drop 1-2 pretzels at a time into the boiling water for 20-30 seconds. Do not boil longer than 30 seconds, or the pretzels may have an unpleasant metallic taste. Use a slotted spatula to lift each pretzel out, shake off excess water, and place on the prepared baking sheet. Sprinkle immediately with coarse salt while still wet. Arrange 6 pretzels on each pan.
Bake. Bake at 425°F until golden brown, about 8-10 minutes.
Brush with butter. Brush the warm pretzels generously with melted butter and serve immediately.