Place warm water in large warm bowl. Sprinkle in yeast; stir until dissolved. Add warm milk, sugar, butter, salt, bran and 1 cup flour. Beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally. Add eggs and 1 cup flour. Beat 2 minutes at high speed. Stir in enough remaining flour to make soft dough. Knead until smooth and elastic, about 8 to 10 minutes. Place dough in lightly greased bowl, turn to grease top. Cover with plastic wrap and refrigerate for future use. Punch dough down whenever doubled in size and use within three days.
Remove dough to lightly floured surface. Divide into 18 equal pieces. Shape into smooth balls. Place balls, about 2 inches apart on large greased baking sheet. Cover; let rise in warm, draft-free place until doubled in size, about 45 minutes.
Bake at 375°F for 11 to 13 minutes or until done. Remove from pans; cool on wire rack.
To bake without refrigerating: After placing dough in lightly greased bowl, cover; let rise in warm, draft-free place until doubled in size, about 45 minutes. Shape dough and proceed as directed.