Combine the first 5 ingredients in the bowl of a stand-up mixer, and let stand 25 minutes.
Dissolve granulated sugar and yeast in warm water; let stand 5 minutes or until foamy. Add to oat mixture. Lightly spoon flours into dry measuring cups; level with a knife. Gradually add 2 3/4 cups all-purpose flour and 3 cups whole wheat flour to oat mixture. Beat at medium speed until well blended. Turn dough out onto a floured surface. Knead until smooth and elastic (about 8 minutes); add enough of the remaining all-purpose 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, 1 hour or until doubled in size. (Gently press two fingers into dough. If indentation remains, dough has risen enough.) Punch dough down; cover and let rest 5 minutes. Divide in half. Working with one portion at a time (cover remaining dough to prevent drying), roll each portion into a 14 x 8-inch rectangle on a floured surface. Roll up each rectangle tightly, starting with a short edge, pressing firmly to eliminate air pockets; pinch seam and ends to seal. Place each loaf, seam sides down, in a 9-inch loaf pan coated with cooking spray. Cover and let rise 30 minutes or until doubled in size.
Preheat oven to 350º.
Uncover dough, and brush egg evenly over loaves. Bake at 350º for 45 minutes or until loaves are browned on bottom and sound hollow when tapped. Remove from pan, and cool on wire racks.