In stand mixer bowl, add 1 cup bread flour, yeast, and salt; mix well. Add warm water and oil to flour mixture. Using a paddle attachment, blend on low speed for 30 seconds. Add egg, honey, vanilla and bananas; beat on medium speed for 3 minutes.
Switch to dough hook attachment. Gradually stir in whole wheat flour and enough remaining bread flour to make a firm dough. Knead until smooth and elastic, about 5 to 7 minutes.(NOTE: For hand mixing: following above steps, mix ingredients in large mixing bowl using a wooden spoon or dough whisk. Knead on floured surface until smooth and elastic.) Place dough in lightly oiled bowl and turn to grease top. Cover; let rise until indentation remains after poking dough with finger down to second knuckle, about 2 hours.
On lightly floured surface, roll or pat dough to a 14x7-inch rectangle. Starting with shorter side, roll up tightly, pressing dough into roll with each turn. Pinch edges and ends to seal. Place in greased 8x4-inch or 9x5-inch bread pan. Cover; let rise until indentation remains when lightly touched.
Preheat oven to 400°F.
Bake for 15 minutes; reduce heat to 375°F and bake 20 to 25 minutes longer until browned and loaves sound hollow when tapped. Bread is thoroughly baked when its internal temperature tests 190°F. Remove from pan; cool on rack.