In a small saucepan, whisk together the flour, milk, and water. Bring the mixture to a simmer, whisking continuously until it thickens, about 6 to 8 minutes. Pour it into a small heatproof bowl, cover with plastic wrap, and set aside to cool slightly.
In a large bowl, combine the flour, sugar, yeast, and salt, and mix well. Add the egg, milk, and cooled starter mixture. Knead the mixture until it comes together, then continue kneading for a few more minutes. Add the butter and knead for another 10 to 15 minutes until the dough becomes uniformly smooth and springy.
Divide the dough into two parts, one slightly larger than the other. Roll the larger portion into a smooth ball and place it in a buttered bowl. Cover the bowl with plastic wrap.
Cut the remaining portion of dough into two pieces, one slightly larger than the other. Incorporate the cocoa powder and 1 tablespoon of milk into the larger piece, kneading until you achieve a uniform light brown color. Form the dough into a smooth ball and place it in a buttered bowl. Cover the bowl with plastic wrap.
Incorporate the dark chocolate cocoa and espresso powders with a teaspoon of milk into the remaining dough ball, kneading well until you achieve a uniform dark brown color. Form the dough into a smooth ball and place it in a buttered bowl. Cover with plastic wrap. Allow all three doughs to rise, covered, until they have doubled in size, about 50 minutes to 1 hour.
Butter a 9x5-inch (23 cm x 12 cm) loaf pan well. Once the dough has doubled in size, remove each dough ball from its bowl one at a time and punch it down. Divide each ball of dough into seven equal parts, forming each part into a smooth ball. Cover all the dough with plastic wrap to keep it moist.
Take a light brown dough ball and roll it out into a small rectangle. Roll the long side of the rectangle tightly to form a log, ensuring that no air is trapped inside. Next, take a dark brown dough ball and roll it into a rectangle, matching the length of the light brown log. Wrap the dark brown rectangle around the light brown log and set it aside. Finally, take one of the lightest dough balls and roll it into a rectangle.
Roll the light and dark brown dough log inside the rectangle, stretching it a bit if necessary to match the length of the pan. Place the log in the pan and repeat with the remaining dough. Once all the dough is rolled, cover the loaf pan with plastic wrap, place a sheet pan on top, and set aside to rise for another 40 to 50 minutes. Halfway through the proof, preheat the oven to 350°F (176°C).
When the dough has risen again, remove the sheet pan and plastic wrap. Brush the top of the loaf with whole milk and bake for 35 to 40 minutes. Once the loaf is done, remove it from the oven and allow it to cool in the pan for about 10 minutes. Then, turn out the loaf and finish cooling on a wire rack for at least one hour before slicing and serving.