Place 1 1/2 cups water, butter, brown sugar, yeast, and salt in the bowl of a stand mixer fitted with the hook attachment. Mix until well combined. Let sit for about 5 minutes.
Turn mixer on low speed and add 4 cups flour, 1 cup at a time. Add additional 1/2 cup flour if dough appears sticky. Increase speed to medium and knead until dough is smooth and pulls away from the sides of the bowl, 4 to 5 minutes. Sprinkle in more flour if dough still seems a little tacky.
Oil a large mixing bowl. Add the ball of dough. Cover with plastic wrap and let rise in a warm place for about 1 hour.
Preheat the oven to 425 degrees F (220 degrees C). Grease a sheet pan with cooking spray.
Bring 3 quarts water to a boil in a large saucepan. Slowly add baking soda and return to a boil.
Meanwhile, remove dough from the bowl, flatten into a disc shape, and cut into 8 wedges. Roll each wedge into a 24-inch rope. Cut each rope into 1-inch pieces.
Drop batches of dough pieces into the boiling water for about 35 seconds. Remove to the prepared sheet pan. Beat egg with 1 tablespoon water; brush over the tops of each boiled dough piece. Repeat with remaining dough.
Bake in the preheated oven until golden brown, 14 to 16 minutes. Cool slightly before serving.