Place water in a large bowl. Sprinkle in yeast; stir until dissolved. Add bran into yeast mixture; let stand 10 minutes.
Add 1 cup flour, sugar, and salt. Beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally. Stir in enough remaining flour to make a soft dough.
Knead on lightly floured surface until smooth and elastic, about 8 to 10 minutes. Place in greased bowl, turning to grease top. Cover; let rise in warm, draft-free place until doubled in size, about 45 minutes.
Punch dough down; turn out onto lightly floured surface. Divide dough into 12 equal pieces. Roll each piece to 13-inch rope; shape into pretzels. Place on greased baking sheet. Brush lightly with beaten egg; sprinkle with salt.