Combine sourdough starter and beer in the bowl of a stand mixer. Add flour, butter, and yeast. Mix until a shaggy dough forms. It will seem very dry and lumpy. Cover and rest 10 minutes.
While the dough rests, mix 2 tablespoon water and 1 teaspoon salt in a small bowl and mix to dissolve the salt.
After 10 minutes, the dough should look slightly more hydrated and less dry. Drizzle the salt water over the dough and use your hand in a pinching motion to squeeze the salt into the dough. Fold the dough over itself a few times to incorporate the salt.
Attach the bowl to a stand mixer fitted with the dough hook. Knead on low until the dough forms a messy ball around the hook, then increase speed to just below medium and knead 4-5 minutes until smooth and soft.
Shape the dough into a ball by folding the dough over itself and tucking the ends under so you have a smooth surface. Place dough back in the bowl, cover, and let rest 45-60 minutes in a warm spot until just about doubled in size.
Preheat oven to 450°F.
When the hour is almost up, whisk together 5 cups of water and ⅓ cup baking soda in a wide, high-sided skillet. Bring to a low boil. Line a sheet pan with parchment paper or a silicone mat and set aside.
Divide dough in 6 equal pieces using a kitchen scale for precision (they should be around 94 grams each). Roll the first piece into a 30" rope. Bend the rope into a U-shape, cross the ends over each other a few times, then fold the twist down and press the ends of each rope down on the bottom curve of the U in a classic pretzel shape. Repeat until all pretzels are shaped.
Boil pretzels in baking soda solution for 45 seconds. Remove to prepared sheet pan.
Brush boiled pretzels with egg wash and top with pretzel salt. Bake 12-14 minutes until deeply browned.
Let cool on sheet pan for 5 minutes, then remove to rack to finish cooling.