Soft pretzels get their signature chewy, brown exterior from being boiled in an alkaline solution--the 1/4 cup baking soda in the water isn't a typo!
In a large bowl, stir together 1 1/2 cups of the bread flour and the yeast; set aside. In a medium saucepan, heat and stir milk, sugar, oil, and the 1 teaspoon salt just until warm (120° to 130°). Add milk mixture to flour mixture. Beat with an electric mixer on medium speed for 30 seconds, scraping sides of bowl constantly. Beat on high speed for 3 minutes. Stir in whole wheat flour and as much of the remaining 1/2 to 1 cup bread flour as you can.
Turn dough out onto a lightly floured surface. Knead in enough of the remaining bread flour to make a moderately stiff dough that is smooth and elastic (6 to 8 minutes total). Shape dough into a ball. Place in a lightly greased bowl, turning once to grease surface. Cover and let rise in a warm place until double in size (about 1 1/4 hours).
Punch dough down. Turn dough out onto a lightly floured surface. Cover and let rest for 10 minutes. Grease two baking sheets; set aside. Divide dough into 12 portions; shape each into a smooth, oval-shaped roll. Place rolls 2 inches apart on prepared baking sheets. Make a crisscross slash across top of each roll with a sharp knife or scissors. Cover; let rise until nearly double (45 to 60 minutes).
Bake in a 475°F oven for 4 minutes. Remove from oven. Reduce oven temperature to 350°F.
Meanwhile, line two large baking sheets with parchment paper; set aside. In a large Dutch oven bring the 3 quarts water to boiling. Gradually stir the baking soda into the boiling water (it will foam up slightly). Carefully lower rolls, two or three at a time, into the water and boil gently for 2 minutes, turning once. Remove with a slotted spoon and let stand on paper towels for a few seconds before placing about 1 inch apart on the prepared baking sheets.
In a small bowl, combine beaten egg white and the 1 tablespoon water. Brush mixture over rolls. Sprinkle with the coarse salt.
Bake in a 350°F oven for 12 to 15 minutes or until golden brown, rotating baking sheets halfway through baking. Immediately remove from baking sheets; cool on wire racks for 10 minutes before serving.