Heat water and milk to a boil; remove from heat. Stir in bran and 1/3 cup butter. Let cool until very warm (120° to 130°F). Mix 2 cups flour, undissolved yeast, sugar and salt in a large mixer bowl. Stir warm liquid into dry ingredients. Mix in eggs and enough additional flour to make a soft dough. Grease top of dough. Cover tightly and refrigerate 2 to 24 hours.
Punch dough down; let stand at room temperature for 15 minutes. Divide dough in half; roll one half to 13 X 12 inches on lightly floured surface. Melt remaining butter; brush on dough. Cut 13 X 2-inch strip from rectangle; reserve. Sprinkle 1/2 cup nuts over remaining dough; cut lengthwise into 5 (2-inch-wide) strips. Stack strips, placing reserved strip, butter-side down, on top of stack. Cut crosswise into 12 equal pieces. Place cut-side up in 12 greased 2-1/2-inch muffin-pan cups. Repeat with remaining dough. Brush tops of rolls with remaining butter.
Cover; let rise in warm, draft-free place until doubled in size, about 45 minutes. Bake at 375°F for 15 to 20 minutes or until golden brown. Remove from pans; cool on wire racks.