In the bowl of a stand mixer fitted with the dough hook, combine 1 1/2 cups warm water, 2 tablespoons light brown sugar, 1 package active dry yeast (2 1/4 teaspoons), and 6 tablespoons unsalted butter, melted. Mix briefly until combined, then let sit 5 minutes to allow the yeast to begin activating.
Add 2 1/2 teaspoons kosher salt and 4 1/2 to 5 cups all-purpose flour to the mixer. Mix on low speed until the flour is incorporated.
Increase the mixer speed to medium and knead until the dough is smooth and begins to pull away from the sides of the bowl, about 3 to 4 minutes. If the dough seems too wet, add additional flour, 1 tablespoon at a time, until it reaches the described consistency.
Turn the dough out onto a clean work surface and knead briefly by hand to form a smooth ball.
Grease a large bowl with oil. Place the dough in the greased bowl, turning once to coat all sides with oil. Cover the bowl with a clean towel or plastic wrap and place in a warm, draft-free spot until the dough doubles in size, about 1 hour.
Preheat the oven to 400°F. Line the bottom of a 9×13-inch baking pan with parchment paper and spray or grease the pan; set aside.
Once the dough has doubled, punch it down and transfer it to a lightly floured surface. Using a sharp knife or bench scraper, divide the dough into 12 equal pieces (about 3 ounces each). Shape each piece into a smooth round roll.
Fill a large pot with enough water so the rolls can float, add 1/3 cup baking soda, and bring to a rapid boil.
Working in batches (about 2 rolls at a time to avoid crowding), carefully lower the rolls into the boiling baking-soda water and boil for 30 seconds. Remove each roll with a slotted spoon, let excess water drip off, and transfer to a wire rack or a lightly floured surface while you boil the remaining rolls.
Place the boiled rolls in the prepared baking pan, arranging them close together. With a very sharp knife or a bread lame, score an X on the top of each roll.
In a small bowl, whisk together 1 large egg and 1 tablespoon cold water. Brush the tops of the rolls with the egg wash, then sprinkle generously with coarse sea salt.
Bake in the preheated 400°F oven for 22 to 27 minutes, or until the rolls are deep golden brown.
Remove the pan from the oven and let the rolls sit for 5 minutes. Serve warm.