Preheat the oven to 400°. Line 2 baking sheets with parchment paper. In a small bowl, whisk together the water and yeast; let stand for 5 minutes.
Using a standing mixer fitted with a dough hook, combine the flour, brown sugar and table salt. Mix in the egg yolk, butter and yeast mixture on low speed until the dough is smooth and elastic, 5 to 7 minutes. Let the dough rest for 5 minutes.
In a large, shallow baking dish, combine the baking soda with 2 cups warm water. On a work surface, divide the dough into
6 equal pieces. Using your hands, roll out each piece into a rope about 42 inches long; dip in the baking soda mixture for 30 seconds. Using the prepared pans as a work surface, shape each rope into a pretzel.
In a small bowl, beat the egg white with 1 tablespoon water. Brush the pretzels with the egg white mixture. Sprinkle with the coarse salt and bake until deep golden, 12 to 15 minutes.