Step 1
Combine flour sourdough starter water powdered milk butter brown sugar yeast and salt in the bowl of a stand mixer fitted with a dough hook. Mix until dough comes together. Knead until smooth and elastic about 5 minutes. Add additional water 1 tablespoon at a time if dough is dry.
Step 2
Remove dough from the bowl and shape into a ball. Clean and dry bowl; grease lightly. Return dough to the bowl. Cover and let rise at room temperature for 1 hour.
Step 3
Turn dough out onto a lightly greased surface. Fold and shape into a rectangle. Divide equally into 12 pieces. Keep pieces covered with plastic wrap while working. Roll out 1 piece at a time into a rope about 18 inches long.
Step 4
Loop and twist 1 rope into a pretzel shape. Place on a baking sheet while you finish rolling out and shaping the remaining pretzels.
Step 5
Preheat the oven to 450 degrees F (230 degrees C).
Step 6
Bring 3 cups water to a boil. Reduce heat to medium-low and stir in baking soda until dissolved. Lower 1 pretzel into the boiling water bath for 30 seconds flipping halfway through. Remove using a large flat spatula and place on a baking sheet. Repeat with remaining pretzels.
Step 7
Beat egg with 1 tablespoon water to make egg wash. Brush each pretzel with egg wash and sprinkle with coarse salt.
Step 8
Bake in the preheated oven until dark golden brown 12 to 15 minutes. Remove from the oven and brush each pretzel with melted butter. Cool for 5 minutes before serving.