Here is the process I used:
3/4 cup warm milk (105–115°F)
1/2 cup plus 2 teaspoons sugar
3 teaspoons active dry yeast
3 1/4 cups all-purpose flour plus additional for dusting
2 whole large eggs
1 egg yolk
1 teaspoon pure vanilla extract
3/4 teaspoon salt
1 1/4 sticks (10 tablespoons) unsalted butter, cut into pieces and softened
For egg wash
1 large egg yolk
1 tablespoon heavy cream or whole milk
For chocolate filling
5 tablespoons unsalted butter, well softened
8oz fine-quality bittersweet chocolate (no more than 60% cacao if marked), finely chopped
1/4 cup sugar
Pour warm milk bread machine and add 2 teaspoons of the sugar and 3 tsp active dry yeast.
Stir with your spoon to mix. Then add flour, whole eggs, yolk, vanilla, salt, and remaining 1/2 cup sugar and start the dough cycle on your bread machine.
Line each loaf pan with 2 pieces of parchment paper (1 lengthwise and 1 crosswise).
Punch down dough with a lightly oiled rubber spatula, then halve dough. Roll out 1 piece of dough on a well-floured surface with a lightly floured rolling pin into an 18- by 10-inch rectangle and arrange with a long side nearest you.
Beat together yolk and cream. Spread 2 1/2 tablespoons softened butter on dough, leaving a 1/2-inch border all around. Brush some of egg wash on long border nearest you.
Sprinkle half of chocolate evenly over buttered dough, then sprinkle with half of sugar (2 tablespoons). Starting with long side farthest from you, roll dough into a snug log, pinching firmly along egg-washed seam to seal. Bring ends of log together to form a ring, pinching to seal. Twist entire ring twice to form a double figure 8 and fit into one of lined loaf pans.
plastic wrap (buttered side down) and let babkas rise in a draft-free place at warm room temperature until dough reaches top of pans, 1 to 2 hours. I sprinkled one with sugar, the other with chocolate.
Put oven rack in middle position and preheat oven to 350°F.
Brush tops of dough with remaining egg wash. Bake until tops are deep golden brown and bottoms sound hollow when tapped (when loaves are removed from pans), about 40 minutes. Transfer loaves to a rack and cool to room temperature.