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 2 half-sheet baking trays with parchment paper or a baking mat (such as a Silpat 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). Divide into 8 equal-size pieces (this doesn't need to be exact). 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 prepared baking sheet. Cover with plastic wrap or a clean kitchen towel while you prepare the next step.
Preheat the oven to 450°F. Bring 10 cups of water and ⅔ cup baking soda to a boil in a large pot.
Place the pretzels in the boiling water, one by one, for 20-30 seconds on each side. A longer boil will create a thicker crust. 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 shake off excess water. Return the pretzels to the baking sheet. Lightly 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 your favorite dipping sauce.