Dissolve yeast in 1 cup warm water in a large bowl; let stand 5 minutes. Lightly spoon flours into dry measuring cups; level with a knife. Add 1 1/2 cups bread flour, whole wheat flour, honey, butter, and salt to yeast mixture; stir until well blended. Add 1/4 cup bread flour; stir until a soft dough forms.
Turn dough out onto a floured surface. Knead until smooth and elastic (about 8 minutes); add enough of remaining bread flour, 1 tablespoon at a time, to prevent dough from sticking to hands (dough will feel tacky).
Place dough in a large bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in size. (Press two fingers into dough. If indentation remains, the dough has risen enough.) Punch dough down; cover and let rest 5 minutes.
Preheat oven to 425°.
Coat 12 muffin cups with cooking spray. Divide dough into 12 equal portions. Working with 1 portion at a time, divide each portion into 3 pieces; shape each piece into a ball (cover remaining dough to prevent it from drying). Place 3 balls in each muffin cup. Cover and let rise 25 minutes or until doubled in size.
Uncover dough. Combine 1 tablespoon water and egg; brush over rolls. Bake at 425° for 12 minutes or until lightly browned. Serve warm.