Combine the warm water (1 cup + 2 Tbsp) and brown sugar in the bowl of a stand mixer. Sprinkle the yeast on top and let sit for 5 minutes or until the yeast begins to foam.
Add the sourdough discard, flour, salt, and butter, and mix using the dough hook attachment until well-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.
Oil a large bowl and transfer the dough in a ball to the bowl. Cover with plastic wrap and let sit for 60-90 minutes, or until doubled in size.
Preheat the oven to 450°F. Line 2 half-sheet baking trays with parchment paper and set aside. Bring 10 cups of water and ⅔ cup baking soda to a boil in a large pot.
In the meantime, turn the dough out onto a smooth, dry work surface (no need to flour or oil, you'll want some traction). Divide into 8 equal pieces. Roll each piece into a 24-inch rope*. 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. Place on the parchment-lined baking pan.
Place the pretzels in the boiling water, one by one, for 20-30 seconds on each side. The pretzels should float (called "passing the float test"); if not, see notes section below. Remove the pretzels from the water using a slotted spatula and return to the baking sheet. Brush each pretzel with egg wash and sprinkle with flaky sea salt**.
Bake until golden brown, approximately 13-15 minutes. Transfer to a cooling rack to cool fully. Serve with a mustard or your favorite dipping sauce.