Make Dough: Heat and stir milk, sugar, butter, and salt in a small saucepan just until warm (110 degrees F to 115 degrees F/43 degrees C to 46 degrees C) and butter is almost melted. Pour 1/4 cup warm water into the bowl of a stand mixer fitted with a paddle attachment. Sprinkle yeast over warm water. Stir to dissolve. Stir remaining milk mixture into yeast mixture.
Add eggs and 2 1/2 cups flour to mixer bowl. Beat at low speed until combined. Beat at high speed for 15 seconds. Switch to dough hook attachment. Add remaining 2 cups flour, 1/2 cup at a time. Mix until dough leaves sides of bowl, about 5 minutes. Transfer dough to a large, lightly greased bowl, turning dough to coat. Cover; let rise in a warm spot until doubled in size, about 1 hour.
Make Filling: Melt 2 tablespoons butter in a large skillet over medium heat. Add apples; cook, stirring frequently, until just beginning to soften, about 4 minutes. Remove from heat. Stir in lemon juice.
Melt 4 tablespoons butter. Transfer to a bowl. Stir in brown sugar, flour, chai spice, and vanilla (mixture will be crumbly). Stir in egg whites.
Shape Babka: Punch dough down. Turn dough out onto a large, lightly floured surface, cover with a bowl, and let rest 10 minutes. Line 3 (4×8-inch) loaf pans with parchment paper, leaving a 2-inch overhang on long sides.
Divide dough into thirds. Roll out each third into a 12-inch square on a lightly floured surface. Spoon one-third of the brown sugar filling (about 2/3 cup) onto each dough square and spread to 1/2 inch from edges. Evenly sprinkle one-third of the apple filling (about 1 heaping cup) over each dough square. Roll up each square tightly to form a log. Pinch seam closed to seal. With your palms, roll logs back and forth to make uniformly cylindrical.
With a sharp knife or dough scraper, cut one log in half lengthwise, exposing striped interior. Lay one half over the other, striped sides out, in an X. Twist ends below cross together 2 or 3 times. Repeat with ends above cross, twisting in opposite direction to form a spiral. Fit into prepared pan, patting down any loose filling and tucking ends under. (It might look like a mess now, but it comes out beautifully!) Repeat with remaining logs and pans. Cover pans with a kitchen towel and let rise in a warm spot until doubled in size, about 1 hour.
BAKE BABKA: Preheat the oven to 350 degrees F (175 degrees C). Melt remaining 2 tablespoons butter; brush over loaf tops. Bake loaves until puffed, well-browned, and an instant-read thermometer inserted into centers registers 200 degrees F (93 degrees C), about 45 minutes, tenting with foil if tops get too brown before loaves are done. Cool in the pans for 10 minutes, then use parchment to lift loaves from pans to transfer to wire racks.
Cool completely on wire racks, about 1 hour, before slicing and serving. (To freeze babka, wrap tightly and freeze up to 2 months. Thaw at room temperature.)
Diy chai spice
Combine 2 teaspoons ground cardamom; 1 teaspoon cinnamon; 1 teaspoon ground ginger; 1 teaspoon ground fennel seeds; 1/4 teaspoon ground cloves; and a dash of black pepper. (Store in an airtight container up to 6 months.)
