In the bowl of a stand mixer, whisk together the flour, granulated sugar, yeast, and salt. Make a well in the center.

In a separate bowl, whisk together the lukewarm milk, eggs, and vanilla extract. Pour the wet ingredients into the well in the dry ingredients.

Attach the dough hook to the stand mixer and mix on low speed until a shaggy dough forms. Gradually add the melted and slightly cooled butter, mixing until fully incorporated. Increase speed to medium-low and knead for 8-10 minutes, until the dough is smooth, elastic, and pulls away from the sides of the bowl. It will be a soft, slightly sticky dough.

Lightly grease a large bowl with butter. Transfer the dough to the bowl, turning to coat. Cover with plastic wrap and let rise in a warm place for 1 to 1 1/2 hours, or until doubled in size.

Once risen, gently punch down the dough. Cover it again with plastic wrap and refrigerate for at least 1 hour, or preferably overnight. Chilling makes the dough much easier to handle.

While the dough is chilling, prepare the filling. In a medium bowl, beat together the room temperature butter and cream cheese with an electric mixer until light and fluffy (about 2-3 minutes).

Gradually add the sifted powdered sugar, mixing on low speed until combined, then increase to medium and beat until smooth. Stir in the vanilla extract.

Gently fold in the rainbow sprinkles. Set aside at room temperature.

Lightly grease a 9x5 inch loaf pan and line with parchment paper, leaving an overhang on the long sides for easy removal. Set aside.

On a lightly floured surface, roll out the chilled dough into a large rectangle, approximately 12x16 inches. Try to keep the edges as straight as possible.

Evenly spread the birthday cake filling over the entire surface of the dough, leaving a small 1/2-inch border along one of the long edges.

Starting from the long edge opposite the border, tightly roll up the dough into a log. Pinch the seam gently to seal.

Trim about 1/2 inch off each end of the log to create neat edges. Using a sharp knife, slice the log lengthwise down the middle, creating two long halves, exposing the swirled filling.

Place one half over the other, forming an 'X' shape. Gently twist the two pieces together, keeping the cut sides facing up as much as possible. It doesn't have to be perfect; the rustic look is charming.

Carefully transfer the twisted dough into the prepared loaf pan. Cover loosely with plastic wrap or a clean kitchen towel and let rise in a warm place for 45-60 minutes, or until visibly puffy.

Preheat your oven to 350°F (175°C) during the last 15 minutes of the second rise.

Bake for 35-40 minutes, or until the top is golden brown and a wooden skewer inserted into the center comes out clean (or with moist crumbs, not wet batter). If the top browns too quickly, tent loosely with aluminum foil.

Remove from the oven and let cool in the pan for 10-15 minutes, then use the parchment paper overhang to lift the babka out onto a wire rack to cool completely.

While the babka cools, whisk together the sifted powdered sugar, 2 tablespoons of milk, and vanilla extract in a small bowl until smooth. Add more milk, 1/2 teaspoon at a time, if a thinner glaze is desired.

Once the babka is completely cool (or at least warm, not hot), drizzle the vanilla glaze generously over the top. Let the glaze set before slicing and serving.