Add the yeast to the lukewarm water and whisk for ~1 minute until combined.
In the bowl of a stand mixer, add the melted butter, brown sugar, salt, and yeast mix. A stand mixer paddle might have trouble at this point with so little in the bowl, so just mix by hand with a spatula until fully combined. From here, with the dough hook attachment, slowly add the first 3 cups of flour while the mixer is on low, scraping down the sides as necessary. Once the dough has thickened and started to come together, add another 3/4 cup of flour. Check the dough at this point, if it's fully combined but still sticky to handle, add the remaining 1/4 cup.
Once the dough is not sticky and has some bounce to it, turn it out onto a lightly floured work surface. Knead the dough for 2 minutes (set a timer, 2 minutes is probably longer than you think to knead by hand!) and then work into a ball. Place the ball in a lightly greased bowl - just use cooking spray. Cover the bowl tightly with plastic wrap and let sit at room temp for 10 minutes.
Preheat the oven to 425°F.
In a large, wide-mouthed pot, add 9 cups of water and the 1/2 cup baking soda to a boil. While the water is heating up, line two baking sheets with parchment paper and set aside.
Once the dough is done resting, cut the dough into 6 equal pieces and roll those smaller pieces into balls, then place 5 back in the greased bowl. Roll the first ball into a long rope. You don't need to be very precise here, just know that they will rise in the oven so don't make the rope too thick. With a sharp blade or a bench scraper, cut the rope into ~1 1/2 inch pieces. I was able to get ~10 pieces per rope, though some were a bit oversized.
Gently place ~10 bites at a time into the boiling water for 20 seconds. Be careful not to drop them in and splash the water around; I set them in the basket of a spider strainer like this one and lowered them into the water. A slotted spoon would also work, though you won't be able to do as many at a time. Once 20 seconds has passed, remove them from the water and place them onto the parchment lined baking sheets, making sure to keep them separated. Repeat for the rest of the dough. Brush all the dough bites with the beaten egg, and sprinkle liberally with the coarse salt. Bake the bites for 12-15 minutes, until golden brown. Make sure to rotate the pans halfway through the bake to make sure they bake evenly.
Enjoy!