1. Begin by warming 1 1/2 cups of water and mixing it with 1 tablespoon of sugar, 2 teaspoons of salt, and 1 packet of active dry yeast in a large bowl. Let this sit for about 5 minutes to activate the yeast.
2. Add 4 1/2 cups of all-purpose flour and 2 ounces of melted unsalted butter to the yeast mixture. Mix until the dough is smooth and consistent.
3. Knead the dough for approximately 4-5 minutes, ensuring it pulls cleanly away from the bowl.
4. Allow the dough to rise in a warm place until it has doubled in size, which takes roughly one hour. Cover the bowl with a clean towel or plastic wrap to keep the dough moist.
5. Once risen, divide the dough into equal pieces. Roll each piece into a rope shape and mold into a pretzel shape.
6. Bring 10 cups of water and 2/3 cup of baking soda to a boil in a large pot. Boil each pretzel for 30 seconds on each side, ensuring they develop the traditional pretzel flavor and chewy texture.
7. Brush the boiled pretzels with a mixture of 1 large egg yolk and 1 tablespoon of water.
8. Sprinkle liberally with pretzel salt or sea salt, depending on your preferred level of saltiness.
9. Heat vegetable oil in a fryer or deep pot to a temperature of 350°F to 365°F.
10. Fry each pretzel donut for about one minute on each side until they are a deep golden brown. Remove from the oil and drain excess oil on paper towels.
11. Prepare the glaze by whisking together 2 cups of powdered sugar, seeds from a vanilla bean (or 2 teaspoons of vanilla extract), and 1/4 cup of whole milk until smooth and well-combined.
12. Drizzle the glaze generously over the warm pretzel donuts or dip the donuts into the glaze for full topping coverage. Allow the glaze to set before serving.