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.

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.

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.

Loop and twist 1 rope into a pretzel shape. Place on a baking sheet while you finish rolling out and shaping the remaining pretzels.

Preheat the oven to 450 degrees F (230 degrees C).

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.

Beat egg with 1 tablespoon water to make egg wash. Brush each pretzel with egg wash and sprinkle with coarse salt.

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.