Combine the warm water (1 cup + 2 Tbsp) and brown sugar in the bowl of a stand mixer fitted with the dough hook attachment*. Sprinkle the yeast on top and let sit for 5 minutes or until the yeast begins to foam.
Add the sourdough discard, flour, and salt and mix on low speed. While the mixer is running, pour in the melted butter and mix until combined. Change to medium speed and continue to knead until a smooth dough forms, about 4-5 minutes (you can also knead by hand if preferred).If needed, add more flour or water (1 Tablespoon at a time) to reach your desired consistency. The dough should be fairly sturdy, smooth and not sticky.
Transfer the dough in a ball to a large, oiled bowl. Cover with plastic wrap and let sit for 60-90 minutes, or until doubled in size.
Line a half-sheet baking pan with parchment paper or a Silicone baking mat and set aside.
Turn the dough out onto a smooth, dry work surface (no need to flour or oil, you'll want some traction). Roll the dough into a long rope, at least 48 inches long. Make a U-shape with the rope, then cross the ends of the rope over each other like a heart. Twist the ends together and push down at the bottom of the U. Transfer to the prepared baking sheet. Cover with plastic wrap or a clean kitchen towel and let rise for 30 minutes.
Preheat the oven to 450°F. Combine 8 cups of water and ½ cup baking soda in a large sautée pan (or another low, wide pan that will fit the size of your pretzel). Stir until dissolved and bring to a boil. Remove from heat and set aside to cool until it's a temperature you can comfortably put your hands into.
Carefully place the pretzel in the cooled baking soda water bath. Using your hands or a large spoon, gently "dunk" the pretzel continuously so it's coated on all sides by the baking soda water for 30-60 seconds.
Scooping under the pretzel**, carefully remove the pretzel from the water bath and transfer to a cooling rack to drain off any excess water. Then, transfer the pretzel back to your parchment paper-lined baking sheet. While wet, reshape the pretzel as needed. Brush the pretzel with the egg wash and sprinkle with flaky sea salt.
Bake until golden brown, approximately 15-17 minutes. Transfer to a cooling rack to cool fully. Serve with your favorite dipping sauces.
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Notes
* You do not need to use a stand mixer to make this recipe. You can also make the dough in a large bowl with a wooden spoon, then knead by hand. **I find it easiest to scoop your hands (and even some of your forearms) under the pretzel in the water bath, then move it to the cooling rack. Give it a minute or two on the cooling rack, then transfer to the baking sheet. You want to avoid excess water pooling around the pretzel on the baking sheet. To make this recipe with active sourdough starter, omit the yeast and replace the sourdough discard with active starter (200g). You'll also need to increase the rise times.