Melt 1/4 cup butter and let it cool to warm/room temperature. Warm 1 1/2 cups whole milk to 107–111°F.

In the bowl of a stand mixer, combine the warm milk, 2 tablespoons brown sugar, and 2 1/4 teaspoons instant yeast. Let sit 5 minutes, until the yeast is bubbly.

Add 2 cups of the 3 1/2 cups bread flour to the bowl. Fit the mixer with the dough hook and mix on low until the flour is mostly incorporated.

Add the cooled melted butter and 1 1/2 teaspoons salt to the dough and mix to combine.

With the mixer running, add the remaining bread flour a little at a time (using the rest of the 3 1/2 cups total; you may add up to 1/2 cup more if needed) until the dough is tacky but does not stick to your hands and the bowl is mostly clean. Knead in the mixer about 5–7 minutes total (until the dough springs back when gently pressed).

Transfer the dough to a well-oiled clean bowl, turn to coat, cover with a tea towel or plastic wrap, and place in a warm spot. Let rise until doubled, about 1 hour.

Meanwhile, prepare a large baking sheet with parchment paper and a light spray or drizzle of oil; set aside.

After the dough has doubled, punch it down, turn it out onto a lightly floured surface, and knead 1–2 minutes. Divide the dough into 8 equal pieces with a bench scraper or knife.

Shape each piece into a round bun and place them on the prepared baking sheet, leaving space between buns. Cover with a tea towel or plastic wrap and let rise in a warm spot until doubled, about 30–60 minutes.

While the buns are undergoing the second rise, heat 4 cups water and 1/4 cup baking soda in a medium pot over high heat. Bring to a gentle boil and then reduce to maintain a gentle simmer.

Preheat the oven to 400°F.

Once the buns have doubled, gently lower each bun into the simmering baking soda bath for up to 5 seconds per side (use a large slotted spoon or spider). Remove and let excess water drip off before returning the buns to the parchment-lined baking sheet.

Using a sharp knife or razor blade, score an X on the top of each bun.

In a small bowl, whisk 1 large egg with a splash of water to make an egg wash. Brush the tops of the buns with the egg wash and, if desired, sprinkle with coarse sea salt.

Bake the buns at 400°F for 10–13 minutes, until deep golden brown.

Remove the buns from the oven. If desired, brush with melted butter. Cool slightly on a wire rack before serving. Store in an airtight container once fully cooled.