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 a little bit foamy.
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, turn the mixer on medium-low speed, and mix for about 5 minutes (until the dough is smooth).
Transfer the dough to a greased bowl. Turn to coat the dough with grease on all sides. Cover and let rest in a warm place until doubled in size, about 1 hour.
Near the end of the rising time, preheat the oven to 425° F. Line two baking sheets with parchment paper. Set aside.
Bring the 9 cups of water and the baking soda to a boil on the stovetop while you shape the dough. Stir to make sure that the baking soda dissolves.
Punch down the dough to release the air. Turn the dough onto a lightly-floured surface. Divide into 12 equal pieces. Roll each piece of dough into a long, thin rope (about 22-24 inches long). Twist each rope into a traditional pretzel shape.
Drop 1-2 pretzels into the boiling water for 20-30 seconds. (Do not boil for longer than 30 seconds, or the pretzels may have an unpleasant metallic taste.) Using a slotted spatula, lift the pretzel out of the water and gently shake to remove as much excess water as possible. Place pretzel onto prepared baking sheet. Sprinkle with coarse salt. Repeat with remaining pretzels, arranging 6 pretzels on each pan.
Bake pretzels in the 425° F oven until golden brown, about 8-10 minutes. Brush the warm pretzels with melted butter and serve immediately.