Make the dough: Mix flour, sourdough starter, water, powdered milk, melted butter, brown sugar, yeast, and salt in a stand mixer fitted with a dough hook until dough comes together. Knead until smooth and elastic, about 5 minutes. Add additional water, 1 tablespoon at a time, if dough is dry.
Remove dough from the bowl and shape into a ball. Clean and dry the bowl, then grease lightly. Return dough to the bowl, cover, and let rise at room temperature for 1 hour.
Turn dough out onto a lightly greased surface. Fold and shape into a rectangle; divide into 12 equal pieces and cover with plastic wrap.
Roll one piece dough into an 18-inch long rope; keep remaining dough covered while you work. Loop and twist the rope into a pretzel shape, and place onto a baking sheet. Roll and shape remaining pretzels.
Preheat the oven to 450 degrees F (230 degrees C).
Prepare a water bath by bringing 3 cups water to a boil in a pot. Reduce the heat to medium-low and stir in baking soda until dissolved. Lower one pretzel into the water bath and cook for 30 seconds, flipping halfway through. Remove with a slotted spoon or spatula and place back onto the baking sheet. Repeat with remaining pretzels.
Beat egg with 1 tablespoon water in a small bowl. Brush egg wash over each pretzel, then sprinkle with coarse salt.
Bake in the preheated oven until dark golden brown, 12 to 15 minutes. Remove from the oven and brush pretzels with melted butter. Cool for 5 minutes before serving.
Editor's Note:
Nutrition data for this recipe includes the full amount of sodium in baking soda. The actual amount consumed will vary.
