Prepare the cream cheese filling by beating softened cream cheese, sugar, and vanilla extract until smooth. Divide into bowls and color with food coloring, then refrigerate.

Make the dough by whisking flour, sugar, and salt. Warm milk, combine with yeast, then allow to foam. Mix in eggs, vanilla, and flour, and knead in butter until smooth. Allow to rise until doubled.

Shape the babka by rolling the dough, spreading the colored filling, rolling it into a log, slicing it, twisting, and placing it in a loaf pan to rise again.

Bake at 350°F for 30-40 minutes until golden brown. Prepare a sugar syrup while baking and brush it onto the warm babka after baking.

Cool the babka in the pan for 10 minutes before transferring to a wire rack to cool completely.