Prepare the Dough: In the bowl of your stand mixer, combine the sourdough starter, warm water, brown sugar, salt, and melted butter. Gradually add in the flour, mixing until a dough forms. Knead the dough for about 4-5 minutes on medium until it becomes smooth and elastic.
Let it Rise: Place the dough in a lightly oiled bowl, cover it with a damp cloth or plastic wrap, and let it rise in a warm place for about 4 hours or until it has doubled in size.
Shape the Pretzels: Once the dough has risen, divide it into 8 equal pieces. Roll each piece into a long rope and shape it into a pretzel. See note below on how to make a classic pretzel shape.
Second rise: Place your pretzels onto your parchment paper or Silpat lined baking sheet and rest for 1 hour.
Boil the Pretzels: Preheat your oven to 450°F (230°C). Bring a large pot of water to a boil and add the baking soda. Carefully drop each pretzel into the boiling water for about 15 seconds on each side, then remove and place them on a baking sheet lined with parchment paper.
Bake to Perfection: Mix your egg and 1 Tablespoon of water together and brush over the boiled pretzels. Sprinkle with coarse salt and bake them in the preheated oven for 12-15 minutes, or until they are golden brown.