In a large mixing bowl, combine the warm milk, sugar, and yeast. Let it sit for about 5 minutes until frothy.
Add the flour, salt, eggs, softened butter, and vanilla extract to the yeast mixture. Mix until a dough forms.
Knead the dough on a floured surface for about 5-7 minutes until smooth and elastic.
Place the dough in a greased bowl, cover with plastic wrap, and let it rise in a warm place for about 1 hour or until doubled in size.
Once risen, punch down the dough and roll it out into a rectangle about ¼ inch thick.
Sprinkle the rainbow sprinkles evenly over the dough. Starting from one long side, roll the dough tightly into a log.
Cut the log in half lengthwise to expose the layers. Twist the two halves together and place in a greased loaf pan.
Cover the pan with plastic wrap and let it rise for another 30 minutes. Preheat the oven to 350°F.
Brush the top of the babka with the beaten egg for a shiny finish. Bake for 30-35 minutes or until golden brown.
Let it cool in the pan for 10 minutes before transferring to a wire rack to cool completely.