In the bowl of a stand mixer, whisk together 1 1/4 cups (159g) bread flour, whole wheat flour, yeast, and salt by hand. Add warm water, 2 tablespoons (28g) melted butter, and honey; using the paddle attachment, beat at low speed until dough comes together. Cover and let stand for 15 minutes.
Add 1/2 cup (64g) bread flour and banana to dough. Switch to the dough hook attachment. Beat at low speed until dough is smooth, elastic, and slightly tacky, 10 to 12 minutes; add up to remaining 1/4 cup (32g) bread flour, 1 tablespoon (8g) at a time, if dough is too sticky. Shape dough into a smooth ball.
Lightly oil a large bowl. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 30 to 45 minutes.
Spray an 8½x4½-inch loaf pan with cooking spray. Line pan with parchment paper, letting excess extend over sides of pan.
Make the filling: In a small bowl, stir together sugar and cinnamon.
Punch down dough, and let stand for 10 to 15 minutes. On a lightly floured surface, roll dough into a 16x7½-inch rectangle. Brush with remaining 2 teaspoons (10 grams) melted butter, leaving a ¼-inch border on one short side. Sprinkle with cinnamon sugar. Starting at short side opposite border, roll up dough, jelly roll style; pinch seam to seal. Place, seam side down, in prepared pan. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 15 to 20 minutes.
Preheat oven to 350°F (180°C).
Make egg wash: In a small bowl, whisk together egg and water. Using a pastry brush, gently brush top of dough with egg wash.
Bake until an instant-read thermometer inserted in center registers 200°F (93°C) to 205°F (96°C), 45 to 50 minutes, covering with foil to prevent excess browning, if necessary. Let cool in pan for 10 minutes. Remove from pan, and let cool completely on a wire rack.