1. Make the Filling: Combine all of the filling ingredients in a medium bowl. Set aside 1 tablespoon of the filling.
2. Make the Dough: Grease a large bowl and set aside. In a 1-cup measuring cup, whisk together the milk, egg yolks and vanilla extract.
3. Using a stand mixer fitted with the dough hook, mix the flour, sugar, yeast and salt on low speed until combined, about 30 seconds. Slowly pour in the milk mixture and mix until the dough comes together, about 3 minutes. Increase the speed to medium-low and add the butter, one piece at a time, until they have all been incorporated, about 1 minute. Continue to mix until the dough is smooth and comes away from the sides of the bowl, 10 to 12 minutes. Transfer the dough to the prepared bowl, cover with plastic wrap, and let rise at room temperature until risen slightly, about 1 hour.
4. Place the bowl in the refrigerator until the dough is firm and has doubled in size, at least 1 hour.
5. Assemble the Babka: Line an 8½ by 4½-inch loaf pan with parchment paper, allowing the excess to hang over the edges.
6. Punch down the dough on a lightly floured work surface. Roll out the dough to a 20 by 14-inch rectangle. Spread all but the 1 tablespoon reserved filling over the dough, leaving a ½-inch border around the edges. Working from the short side, roll the dough into a cylinder and pinch along the seam to seal.
7. Position the cylinder seam side up and roll back and forth until stretched to 18 inches long. Spread the reserved filling over the top of the cylinder. Fold the cylinder on top of itself and pinch the ends to seal.
8. Gently twist the double cylinder twice to form a double figure eight. Place the shaped dough seam side down in the prepared pan, cover loosely with plastic wrap and let rise at room temperature until it doubles in size, about 1 hour.
9. Bake the Babka: While the dough is rising, preheat the oven to 350 degrees F. Lightly beat the whole egg and brush the top of the loaf. Bake until the loaf is deep golden brown and the inside of the loaf registers 190 degrees F on an instant-read thermometer, about 45 minutes. Allow the bread to cool in the pan for 20 minutes, then remove the loaf from the pan and cool completely, about 2 hours. The bread can be kept at room temperature, wrapped tightly in plastic wrap, for up to 3 days.