Dissolve sugar and yeast in 1 cup warm water in a large bowl; let stand 5 minutes. Stir in oil. Weigh or lightly spoon flours into dry measuring cups; level with a knife. Add 9.5 ounces (about 2 cups) bread flour, whole wheat flour, and salt to yeast mixture; stir until a soft dough forms. Turn dough out onto a lightly floured surface. Knead until dough is smooth and elastic (about 5 minutes); add enough of remaining bread flour, 1 tablespoon at a time, to prevent dough from sticking to hands (dough will feel sticky).
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, for 1 hour or until doubled in size. (Gently press two fingers into dough. If indentation remains, the dough has risen enough.) Punch dough down; knead in walnuts. Shape dough into a 9-inch round on a lightly floured surface. Place dough on a large baking sheet sprinkled with cornmeal. Cover and let rise 1 hour or until doubled in size.
Preheat oven to 350°.
Uncover dough. Combine milk and egg white; brush over dough. Score dough by making two diagonal slits with a sharp knife; make two diagonal slits in the opposite direction to create a crosshatch pattern. Bake at 350° for 30 minutes or until bread is browned on bottom and sounds hollow when tapped. Cool on a wire rack.