For light dough: In the bowl of a stand mixer, whisk together 2 1/3 cups (292 grams) all-purpose flour, rye flour, salt, yeast, and caraway seeds by hand until well combined. Add 1 cup (240 grams) warm water, oil, and molasses; using the paddle attachment, beat at low speed until combined, about 30 seconds, stopping to scrape sides of bowl.

Switch to the dough hook attachment. Beat at low speed until dough is soft, tacky, and elastic and mostly cleans sides of bowl, 8 to 12 minutes, stopping to scrape sides of bowl and dough hook; add up to remaining 1/3 cup (42 grams) all-purpose flour, 1 tablespoon (8 grams) at a time, if needed. (Dough may still stick to bottom of bowl but should pass the windowpane test; see Note.) Transfer dough to a clean surface. Cover with a lightly greased sheet of plastic wrap.

For dark dough: Clean bowl of stand mixer and attachments. In the bowl of a stand mixer, whisk together 2 1/4 cups (281 grams) all-purpose flour, rye flour, cocoa, salt, yeast, and caraway seeds by hand until well combined. Add 1 cup (240 grams) warm water, oil, and molasses; using the paddle attachment, beat at low speed until combined, about 30 seconds, stopping to scrape sides of bowl.

Switch to the dough hook attachment. Beat at low speed until dough is soft, tacky, and elastic and mostly cleans sides of bowl, 8 to 12 minutes, stopping to scrape sides of bowl and dough hook; add up to remaining 1/4 cup (31 grams) all-purpose flour, 1 tablespoon (8 grams) at a time, if needed. (Dough may still stick to bottom of bowl but should pass the windowpane test; see Note.)

On a clean surface, shape light and dark doughs into balls. 6. Lightly oil 2 large bowls. Place doughs in bowls, turning to grease tops. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 40 minutes to 1 hour.

Punch down doughs; cover and let stand for 10 minutes. Turn out doughs onto a clean surface. Divide each dough into 4 portions (about 178 grams each). For one loaf, roll 2 light dough portions and 2 dark dough portions into 9x6-inch rectangles. Layer rectangles on top of one another, starting with light and ending with dark. Reroll or pat stacked dough into a 9x6-inch rectangle, making sure to get out any air bubbles between layers. Starting at one long side, roll up dough, jelly roll style. Repeat procedure with remaining dough.

Spray 2 (9x5-inch) loaf pans with cooking spray. Place dough logs, seam side down, in prepared pans. (Alternatively, pinch ends to seal, and tuck under to secure; roll dough into about a 9 1/2 -inch log [about 3 inches wide]. Repeat procedure with remaining dough, and place dough logs 3 inches apart on a parchment paper-lined baking sheet.) Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 40 minutes to 1 hour.

Preheat oven to 375°F (190°C).

For topping: In a small bowl, whisk together egg and

tablespoon (15 grams) water. Brush loaves with egg wash. Sprinkle with caraway seeds. Using a small sharp serrated knife, make 4 diagonal slashes that are 1/2 to 3/8 inch deep and 4 1/2 to 5 inches long across top of loaves.

Bake for 15 minutes; rotate pans, and bake until golden brown and an instant-read thermometer inserted in center registers 190°F (88°C), 10 to 12 minutes more. Remove from pans, and let cool completely on a wire rack. Store in an airtight container for up to 3 days.