In the bowl of a stand mixer, whisk together 1¼ cups (159 grams) bread flour, whole wheat flour, yeast, and salt by hand. Add ¾ cup (180 grams) warm water, 2 tablespoons (28 grams) melted butter, and honey; using the paddle attachment, beat at low speed until dough comes together. Cover and let stand for 15 minutes.

Add ½ cup (64 grams) 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 ¼ cup (32 grams) bread flour, 1 tablespoon (8 grams) 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.

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).

In a small bowl, whisk together egg and 1 tablespoon (15 grams) 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.