Combine the warm water (1 cup + 2 Tablespoons) 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 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 soft, 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 onto a smooth, dry surface. Divide into 8-10 equal-sized pieces (8 for larger buns, 10 for smaller buns). Using one palm for friction, roll the dough into a ball with the seam on the bottom**. Place the buns on the prepared baking sheets and cover with plastic wrap or a clean kitchen towel to rise for 20 minutes while you prepare the next step.
Preheat the oven to 425℉. Add ⅔ cup baking soda to 10 cups of water in a large pot and bring to a boil.
Place the buns in the boiling water for 20-30 seconds on each side. You can likely boil several buns at once, but make sure they have room to float freely. A longer boil will create a thicker crust. The buns should float (called "passing the float test"); if not, see notes section below. Remove the buns from the water using a slotted spatula and shake off excess water. Return the buns to the baking sheet. Lightly brush each buns with egg wash and sprinkle with flaky sea salt. Use a bread scoring tool, sharp knife, or scissors to score an X or a line on top of each bun. This does not need to be deep - just enough to break the surface to allow steam to esacpe. It will expand as the buns bake.
Bake for 14-16 minutes until a deep, golden brown (that classic pretzel color). Remove from the oven and let cool fully before serving.