Combine cider and dried apples in a measuring cup. Microwave at HIGH 1 1/2 minutes. Stir in shredded apple, and set aside.
Combine yeast and warm water; let stand 5 minutes.
Combine apple mixture, yeast mixture, 2 3/4 cups bread flour, and next 5 ingredients in a large bowl; beat at medium speed of an electric mixer until ingredients are well blended. Gradually stir in egg and enough rye flour to make a soft dough.
Sprinkle 2 tablespoons bread flour evenly over work surface. Turn dough out onto floured surface, and knead until smooth and elastic (about 5 minutes). Place in a bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 55 minutes or until doubled in bulk.
Punch dough down; turn out onto work surface, and flatten slightly. Sprinkle walnuts over dough; knead gently to incorporate walnuts. Shape into a 7-inch round loaf. Place loaf on a baking sheet coated with cooking spray. Cover and let rise in a warm place, free from drafts, 40 minutes or until doubled in bulk. With a sharp knife, make 4 crisscross slashes (1/4-inch deep) across top of loaf. Brush loaf with egg white. Bake at 350° for 35 minutes or until loaf sounds hollow when tapped. Remove bread from baking sheet immediately; let cool on a wire rack. Cut into wedges.