Make the Dough: Place the all-purpose flour, granulated sugar, instant yeast, and salt in the bowl of an electric mixer fitted with a dough hook. Stir on medium-low speed for a minute to combine. Add the egg, whole milk, vanilla, and unsalted butter and mix on medium speed until a smooth dough forms, about five minutes. Form into a ball and place in a lightly oiled bowl. Cover with plastic wrap and place in a draft-free area until doubled in volume, about one hour.
Make the Apricot Filling: Combine the dried apricots, orange juice, lemon juice, and granulated sugar in a small saucepan and bring to a simmer over medium-low heat. Simmer uncovered for 10 to 15 minutes, stirring occasionally, until the apricots soften and the liquid is reduced by half. Remove from heat and allow to cool completely. Transfer the mixture to a food processor and process until a puree forms.
Make the Cream Cheese Filling: In a medium bowl, stir the cream cheese and granulated sugar with a wooden spoon until smooth and lump-free. Add the egg yolk, vanilla extract, and salt, and stir to combine.
Make the Streusel: In a medium bowl, whisk together the all-purpose flour, granulated sugar, light brown sugar, and salt. Add the unsalted butter and mix with a rubber spatula until the mixture comes together into large, coarse crumbs. Cover and refrigerate until ready to use.
Assemble the Babka: Lightly grease a 9x5-inch loaf pan and set aside. On a lightly floured surface, roll the dough out into a 10x24-inch rectangle, with the long edge facing you. Spread the apricot filling evenly over the dough, leaving a half-inch border along all of the edges. Spread the cream cheese filling over the apricot filling.
Starting with the bottom edge, roll the dough into the middle of the rectangle, then do the same with the top edge so that the two rolls meet in the center. Visualize the long cylinder divided into three equal lengths. Fold the left third over onto the middle third. Then, fold the right third over the middle third. Pick up the dough and turn it over so the seam is on the bottom. Holding each end, gently twist it in the middle and place it in the prepared pan.
Whisk together the egg and salt for the egg wash and brush it over the babka. Lightly cover the pan with plastic wrap and place it in a draft-free area until it is doubled in size, 1½ to 2 hours.
Preheat oven to 375 degrees F. Brush the babka once again with the egg wash and sprinkle the streusel evenly over the top, pressing lightly so the crumbs adhere. Bake until the top is a deep golden brown, about 50 minutes. Cool the bread in the pan set on a wire rack for 25 minutes, then unmold the bread from the pan and place on the wire rack to cool completely.
