Combine milk, 1 teaspoon sugar, and yeast. Let proof for 5 minutes until foamy.
¼ cup warm whole milk, 1 ½ teaspoons active dry yeast, ¼ cup (50g) granulated sugar
Add flour, sugar, salt, eggs, egg yolks, and milk mixture. Mix until combined. Knead on low speed with dough hook for 3 minutes until smooth or by hand for about 6 minutes. Add butter a piece at a time and keep kneading, then increase the speed to medium and knead for another 3 minutes until smooth. The dough will be sticky. Transfer to clean bowl and let rise until doubled in size, about 2 hours. Punch down down, cover, and transfer to the fridge. Let rise another 4 hours or overnight.
2 ¼ cups (317g) all-purpose flour, ½ teaspoon salt, 2 large eggs + 2 egg yolks, 1 teaspoon pure vanilla extract, 4 Tablespoons salted butter,
For the filling, melt butter and chocolate. Whisk in sugar, cocoa powder, vanilla, and salt. Cool until spreadable, about 25 minutes.
6 Tablespoons salted butter, ½ cup bittersweet or semisweet chocolate,, ½ cup (100g) granulated sugar, ¼ cup natural unsweetened or dutch process cocoa powder, ½ teaspoon salt, 2 teaspoons pure vanilla extract
Line a 8½ by 4½-inch loaf pan with a parchment paper sling. If you only have a 9 by 5-inch pan, you might need to start checking to see if your babka is done 10 to 15 minutes earlier.
Roll out into a 16x12-inch rectangle. Spread with filling. Roll up from short ends (opposite of cinnamon roll approach) and pinch to seal. Cut in half lengthwise. Turn cut sides up, pinch together at one end, and twist other ends. Transfer to a loaf pan with cut sides up. Cover with plastic wrap and let it rise for about 45 minutes, just until it reaches the edges of the pan.
Preheat oven to 350°F. Whisk egg and cream. Brush the top of the loaf with the egg wash, then bake for 35 to 45 minutes until golden brown and the babka has an internal temperature of 190°F.
Egg wash
While the babka is baking, combing the sugar and water for the syrup in a small saucepan over medium heat. Cook for about 5 minutes until the sugar has dissolved completely, stirring occasionally, then set aside to cool.
½ cup (100g) granulated sugar, ¼ cup water
Poke holes all over the warm loaf of babka when it comes out of the oven. Pour the syrup over the warm loaf. Let cool for 20 minutes to 1 hour before removing from the pan. Cool completely before slicing into 1-inch thick slices to serve.