In a bowl, combine sugar, lukewarm milk lukewarm water, white sugar, and yeast. Let it rest until it becomes bubbly.
Once bubbly, add the remaining ingredients for the dough (flour, unsalted butter, egg, salt, and vanilla extract). Mix and knead them together until you have a soft, smooth dough.
Place the dough in a greased bowl, turn it over, and cover it with a damp towel. Allow the dough to rise, covered, for 45 minutes to 1 hour or until it becomes puffy (it may not necessarily double in size).
Making the Filling:
While the dough is rising, prepare the filling. Mix all the filling ingredients (cream cheese, powdered sugar, egg, flour, and vanilla extract) together until the mixture is smooth. Chill the filling until you're ready to use it.
Assembling the Bread:
Transfer the risen dough to a lightly floured surface, punch it down, and let it rest for 5 minutes.
Roll out the dough into a 16 x 12-inch rectangle.
Transfer the rolled dough to a baking pan lined with parchment paper.
Spread the prepared cheese filling lengthwise down the center third of the dough.
Cut 1 to 3-inch-wide strips from each side of the filling out to the edges of the dough.
Fold about an inch of dough at each end over the filling to encase it, then fold the strips, at an angle, across the filling, alternating from side to side.
Baking and Serving:
Allow the braids to rise, covered, for 1 to 1 1/2 hours, until they almost double in size.
Brush the braids with a glaze made from 1 egg yolk beaten with 1 tablespoon of water.
Bake the braids in a preheated 350°F oven for 35 to 40 minutes, or until they turn golden brown.
Remove from the oven, and cool the bread on a wire rack.