Preheat oven to 350 degrees. Grease an 8” x 8” square cake pan.
In a medium bowl, combine cream cheese, cake mix, melted butter, and a pinch of salt. Mix until smooth. Divide into 6 small bowls and tint each one with a different color.
Place bread dough on a lightly floured work surface. Using a rolling pin, shape the dough into a rectangle, ⅛-inch thick and 13” x 10”.
Spread each rainbow filling across dough in separate stripes, length-wise, getting as close to the edges as possible. Starting with the long, red side of the rectangle, roll the dough up into a log.
Use a sharp knife or serrated bread knife to carefully slice the log into 9 equal portions. Transfer to prepared cake pan, placing rolls cut side up, 3-by-3. Transfer to oven and bake until golden and puffed, about 30 minutes. Let cool 10 minutes.
Meanwhile, make glaze. Add confectioners’ sugar, milk, and vanilla to a large bowl. Whisk to combine, adding more milk if necessary to reach a consistency that is thick but pourable.
Drizzle glaze immediately on cooled birthday cake rolls and top with rainbow sprinkles. Serve warm.