Prepare the Dough: In a large bowl, combine the warm milk, warm water, and 1 tablespoon of sugar. Sprinkle the yeast over the mixture and let it sit for about 5-10 minutes until it becomes frothy.

Mix the Ingredients: In a separate large bowl, whisk together the flour, remaining sugar, and salt. Add the yeast mixture, softened butter, eggs, and vanilla extract. Mix everything until the dough begins to come together.

Knead the Dough: Turn the dough onto a floured surface and knead for 5-8 minutes, or until the dough is smooth and elastic. If the dough is too sticky, add a little more flour. Place the dough in a greased bowl, cover it with a clean towel, and let it rise in a warm place for 1 to 1.5 hours, or until it has doubled in size.

Shape the Bread: Punch down the dough and divide it into 3 equal portions. Roll each portion into a long rope (about 12-15 inches long). Twist the ropes together into a braid and shape it into a circle. Pinch the ends together to form a round loaf.

Add the Dyed Eggs: Gently press the hard-boiled, dyed eggs into the dough. You can place them symmetrically around the braid or in any pattern you prefer. Make sure the eggs are securely nestled into the dough.

Let the Dough Rise Again: Cover the braided loaf with a clean towel and let it rise again for about 30-45 minutes, or until it has puffed up.

Prepare the Egg Wash: In a small bowl, whisk together the egg yolk with a tablespoon of water. Brush this mixture over the entire loaf to give it a golden color when baking.

Bake the Bread: Preheat your oven to 350°F (175°C). Place the bread on a parchment-lined baking sheet and bake for 25-30 minutes, or until the bread is golden brown and sounds hollow when tapped on the bottom.

Cool and Serve: Allow the bread to cool on a wire rack. Once cooled, slice the bread and serve. The dyed eggs can be eaten along with the bread or kept as a festive decoration.