Step 1

Scald the milk: Heat the milk in a saucepan set over medium-low heat until tiny bubbles form at the edge of the pan, just before it simmers. An instant-read thermometer will read between 180°F and 185°F. (If you don't have a thermometer, test it with your finger; it should be warm, not uncomfortably hot.) Do not boil. Let cool to room temperature.

Step 2

Dissolve the yeast: In the bowl of a stand mixer fitted with a paddle attachment, combine the warm milk, 2 tablespoons of the sugar, 1/2 cup of the flour, and the yeast. Mix at low speed just until combined, about 20 to 40 seconds. Let stand in the bowl until the mixture is foamy, frothy, smells distinctly like yeast, and is beige in color, 7 to 8 minutes.

Step 3

Make the dough: Add the remaining sugar, powdered milk, and salt. Mix well at medium speed. Slit the vanilla bean pod lengthwise and scrape out the seeds into the mixer bowl. Add 2 of the eggs and the egg yolks and mix well.

Step 4

Add the flour: Switch to a dough hook and add the flour, 1 cup at a time, mixing after each addition at low to medium-low speed until combined. Knead for 5 more minutes.

Step 5

Add the butter: Divide the butter into 2 or 3 parts. Add one at a time and mix for about a minute after each addition, until the butter is incorporated into the dough. Mix another minute, for a total of 3 to 4 minutes. It will be a sticky dough at first, but it will become a smooth, elastic ball that clears the side of the bowl and sticks at the bottom a bit.

Step 6

The first rise: Scrape the dough into a large mixing bowl. Spray nonstick vegetable oil spray over a piece of plastic wrap at least 2 inches wider than the bowl and drape it over the bowl. Cover with a large kitchen towel or two and set aside in a warm but not hot (between 70°F and 80°F) undisturbed spot in the kitchen. Let rise for 2 to 2 1/2 hours, until it has doubled in size. (If you wish, you can let the dough rise in the refrigerator overnight, but allow time for it to come back to room temperature before you bake.)

Step 7

Prepare the pans: Spray 2 bread pans with nonstick vegetable oil spray. Cut 4 sheets of parchment paper; make two big enough to cover the bottom and sides of the pan lengthwise, with an overhang of twice the height of the sides. Make the other two sheets big enough to cover the bottom and sides width-wise, with an overhang of twice the height of the sides. Line each pan with one sheet of each size, placing one lengthwise and the other crosswise, so that the bottom and sides are aligned and there is an overhang of parchment on all sides for easy removal.

Step 8

Prepare the filling: Slit the vanilla bean pod lengthwise and scrape out the seeds into the bowl of a food processor. Add the toasted walnuts and process in 3 (10- to 15-second) pulses. The walnuts will be broken up, not puréed, with some distinct pieces and some walnut powder. Add the sugars, cinnamon, cardamom, and salt; process in 2 (5-second) pulses. Add the egg whites; process in 2 (15-second) pulses. Add the butter; process for 15 to 20 seconds. Scrape into a bowl and set aside.

Step 9

Cut and roll the dough: Once the dough has risen, lightly flour a work surface. Place the dough on it and divide it in half. Working with one piece at a time, roll each into an 8-inch by 14-inch rectangle.

Step 10

Make an egg wash: Whisk together the egg and water in small bowl. Brush the dough lightly with the some of egg wash, leaving 1/2-inch border around the edge uncovered, so you can hold it securely.

Step 11

Preheat the oven: Preheat the oven to 325°F.

Step 12

Spread the filling on the dough: Use a silicone spatula to scoop equal portions of the filling onto each of the rectangles of dough and spread it evenly on each, leaving a 1/2-inch border around the edges for gripping and rolling ease.

Step 13

Roll up the dough: Use a pastry scraper or a spatula to help you lift up the long (14-inch) edge of one piece of dough and roll it up, jelly-roll-style, as tightly as you can into a tube. Pinch the seam at the end, pressing it gently. Place seam-side down, and roll gently back and forth (toward you and away from you) until the tube is 16 to 18 inches long.

Step 14

The second rise in the pan: Lift up one portion of the rolled, filled dough and twist it gently. Fold in half and place it into one of the prepared pans. Repeat with the other roll of dough. Spray 2 pieces of plastic wrap with nonstick vegetable oil spray then cover the babkas. Let the dough rise in the pan for 1 1/2 hours in a warm, draft-free spot, until they are doubled in size.

Step 15

Bake the babkas: Use a pastry brush to brush the tops of the babkas with the remaining egg wash. Bake for 45 to 55 minutes, until the tops are firm, they are golden-brown in color and a cake tester or toothpick inserted into the center of a babka comes out without any dough on it. There will be filling on it — that's fine!

Step 16

Cool in the pans: Cool in the pans for 5 to 10 minutes. Then, using the paper overhangs as handles, carefully lift the babkas out of the pans and let cool on a cooling rack.