In a small saucepan, heat 1 ¼ cups water to 110 degrees. Stir in 1 teaspoon sugar until dissolved and remove from heat. Pour yeast over the top and set aside to bloom until foamy, about 5 to 10 minutes.
Meanwhile, in the bowl of a standing mixer with the dough hook attached, combine flour, salt, and remaining 2 tablespoons sugar.
Add melted butter and bloomed yeast mixture. Stir until the dough just comes together, pulling away from the sides of the bowl but clinging to the bottom.
Coat a large bowl lightly with oil or nonstick spray. Transfer dough to the oiled bowl and cover loosely with plastic wrap coated with nonstick spray. Let rise in a warm place until doubled in size, about 1 hour (see note 2).
Preheat oven to 450 degrees. Line 2 baking sheets with parchment paper or a silicone mat. In a 3-quart saucepan, bring remaining 2 quarts (8 cups) water and baking soda to a boil. In a small bowl, whisk together yolk and water.
Turn out dough on to a lightly floured surface and shape into an 8-inch round disc. Using a bench scraper, pizza cutter, or butter knife, cut the disc into quarters. Then, cut each quarter into half so you have 8 pieces.
Shape each piece of dough into a long rope and lay each rope in a horseshoe shape. Twist the ends together once or twice, then bring the ends towards you. Separate the ends and fold them down, using water or milk to make them stick to the dough. Repeat with remaining pieces of dough.
Working with one pretzel at a time, drop a pretzel in the boiling baking soda water for 30 seconds. Remove and transfer to a prepared baking sheet. Immediately brush with the egg yolk wash and sprinkle with coarse salt. Repeat with remaining pretzels.
Bakeuntil the pretzels are dark brown, crackly, and fragrant, about 8 to 10 minutes. Remove from the oven and serve with Homemade Cheese Sauce.