How to Make It
Prepare the Dough: Whisk together milk, yeast, and 1 cup of the flour in a medium bowl until well combined. Cover with a damp clean towel, and let stand until puffed and airy, about 20 minutes.
Beat butter and granulated sugar with a heavy-duty electric stand mixer fitted with the paddle attachment on medium until very pale and fluffy, about 4 minutes. Add eggs, 1 at a time, beating until well combined after each addition. Add egg yolk, vanilla, and salt, beating until combined, about 1 minute. Add milk mixture and remaining 3 3/4 cups flour, and beat on low speed until wide strands of dough pull away from bowl and color is even throughout, about 2 minutes. Cover tightly with plastic wrap, and chill 8 hours or overnight.
Grease 2 (9- x 5-inch) loaf pans with shortening; set aside.
Prepare the Filling: Stir together all Filling ingredients in a medium bowl. Divide mixture in half; set aside.
Assemble the Babka: Divide Dough in half. Place half in refrigerator until ready to use. Place remaining half on a well floured surface, and roll into a 20- x 12-inch rectangle (about 1/8 inch thick). Spread three-fourths of half of the Filling over Dough, leaving a 1/2-inch border. Starting with 1 long side, tightly roll up Dough, jelly-roll style. Cut log in half crosswise. Spread remaining one-fourth of half of the Filling over 1 log half. Place second log half on top to form an “X” shape, and twist logs together. Place in 1 prepared loaf pan. Cover with plastic wrap, and repeat with remaining Dough half and remaining half Filling. Place loaves in a warm (80°F to 85°F) place, and let stand until doubled in size, 45 minutes to 1 hour.
Preheat oven to 325°F. Whisk together egg and water in a small bowl. Uncover loaves, and brush evenly with egg mixture. Bake in preheated oven until top is deep brown and a thermometer inserted into center of babka registers between 190°F and 200°F, 35 to 40 minutes. Cool completely in pans on a wire rack before slicing, about 1 hour.