In a large mixing bowl, combine warm milk, sugar, and yeast. Stir gently and let it sit for 5-10 minutes until foamy, indicating the yeast is active.

Add the melted butter, egg, salt, and 2 cups of flour to the yeast mixture. Mix with a wooden spoon or a stand mixer with a dough hook until well combined.

Gradually add the remaining 1 cup of flour, mixing until a shaggy dough forms. Turn the dough out onto a lightly floured surface and knead for 8-10 minutes, or until smooth and elastic. Alternatively, knead in a stand mixer for 5-7 minutes.

Place the dough in a lightly oiled bowl, turning once to coat. Cover with plastic wrap or a clean kitchen towel and let it rise in a warm place for 1 to 1.5 hours, or until doubled in size.

For the Beetroot Filling: In a food processor, combine the cooked beetroot, minced garlic, olive oil, chopped dill, salt, and pepper. Pulse until coarsely chopped, but not a smooth paste. Transfer to a bowl and stir in the crumbled feta cheese. Set aside.

For the Spinach Filling: Heat a small saucepan over medium heat. Add the fresh spinach and cook until wilted, about 2-3 minutes. Drain very well, pressing out as much excess liquid as possible. Finely chop the spinach. In a bowl, combine the chopped spinach, softened cream cheese, grated Parmesan, nutmeg, salt, and pepper. Mix until well combined. Set aside.

Punch down the risen dough and turn it out onto a lightly floured surface. Roll the dough into a large rectangle, approximately 18x12 inches (45x30 cm).

Spread the beetroot filling evenly over one half of the dough rectangle (lengthwise). Spread the spinach filling evenly over the other half of the dough rectangle, ensuring the two fillings meet in the middle without mixing too much.

Carefully roll up the dough tightly from one of the long sides, creating a log with the fillings inside. Pinch the seam to seal.

Using a sharp knife, slice the log in half lengthwise, exposing the colorful layers of filling.

Place the two halves next to each other, cut-side up. Gently twist the two halves together, forming a braid. Pinch the ends together to seal.

Carefully transfer the braided bread to a baking sheet lined with parchment paper.

Cover the braided bread lightly with plastic wrap or a kitchen towel and let it rise in a warm place for another 30-45 minutes, or until noticeably puffed up.

Preheat your oven to 375°F (190°C).

For the Egg Wash: In a small bowl, whisk together the egg and milk/water. Gently brush the top and sides of the risen bread braid with the egg wash.

Bake for 30-35 minutes, or until the bread is golden brown and sounds hollow when tapped on the bottom. If the top starts to brown too quickly, loosely tent with aluminum foil.

Remove from the oven and let cool on the baking sheet for 10-15 minutes before transferring to a wire rack to cool completely. Slice and serve warm or at room temperature.