Stir together 1 1/4 cups warm water and sugar in the bowl of a stand mixer. Sprinkle yeast over mixture, and let stand until mixture is foamy, 5 minutes. Fit mixer with paddle attachment. Add flours, 1 tablespoon of the salt, and 1 egg to yeast mixture, and beat at low speed until well combined, about 30 seconds. Replace paddle with dough hook, and beat on medium speed until dough is smooth and has pulled away from the bowl, 5 minutes.
Transfer dough to a lightly greased bowl, turning to grease top. Cover and let rise in a warm place until doubled in bulk, 1 hour.
Preheat oven to 450°F. Lightly grease 2 wire racks, placing each in a rimmed baking sheet. Bring 10 cups of the water to a boil over high.
Meanwhile, turn dough out onto a lightly greased surface; roll out to a 12- x 8-inch (about 3/4-inch-thick) rectangle. Cut into 8 (12-inch-long) strips; cut each strip into 4 pieces.
To shape the pretzels, form 1 strip into a "U" with the ends of the strip pointing up. Fold one end down, making a small loop. Fold the other end down, making a second small loop. Press the dough in place to secure. Repeat with remaining dough.
Add baking soda to water; stir to dissolve. Add 3 to 4 pretzels and cook 3 minutes, turning occasionally. Transfer to prepared pans. Repeat with remaining pretzels.
Let pretzels rest until moisture evaporates, 15 minutes. Whisk together remaining egg and water. Brush egg wash over tops of pretzels, and sprinkle with remaining 2 tablespoons salt.
Bake until tops are golden, 7 to 10 minutes, rotating pans from top to bottom rack about halfway through bake time. Turn pretzels over and bake until tops are golden, 3 to 5 minutes.