1. Combine the Dough Ingredients: In a large mixing bowl, combine the active sourdough starter, warm water, sugar, and salt. Stir until well combined. Gradually add the all-purpose flour, mixing with a wooden spoon or your hands until a shaggy dough forms.

2. Knead the Dough: Turn the dough out onto a lightly floured surface. Knead for about 8-10 minutes, or until the dough is smooth and elastic. It should spring back slightly when poked. Incorporate the melted butter during the last few minutes of kneading.

3. First Rise (Bulk Fermentation): Place the dough in a lightly oiled bowl, turning once to coat. Cover the bowl with plastic wrap or a clean kitchen towel and let it rise in a warm place for 1.5 to 2 hours, or until it has visibly increased in size (it may not double, but should be puffy).

4. Shape the Pretzel Bats: Gently punch down the risen dough and divide it into 12 equal pieces. Roll each piece into a long rope, about 18-20 inches long. To form a bat, create a loop in the middle of the rope, then cross the ends over each other at the top to form the “wings.” Pinch the ends to create pointed tips for the wings and a small head at the top of the loop. Arrange the shaped bats on parchment-lined baking sheets.

5. Prepare the Baking Soda Bath: Preheat your oven to 425 F (220 C). In a large pot, bring 10 cups of water to a rolling boil. Carefully add the 1/2 cup of baking soda to the boiling water (it will foam up, so be cautious).

6. Boil and Bake: Using a slotted spoon or spider, carefully drop one or two pretzel bats into the boiling baking soda solution for 20-30 seconds per side. Do not overcrowd the pot. Remove the pretzels and place them back on the parchment-lined baking sheets.

7. Egg Wash and Salt: Brush each boiled pretzel bat with the beaten egg wash and generously sprinkle with coarse sea salt.

8. Bake to Golden Perfection: Bake for 15-18 minutes, or until the pretzel bats are deep golden brown. Transfer to a wire rack to cool slightly before serving.