Stir together warm milk and 2 teaspoons sugar in bowl of mixer. Sprinkle yeast over mixture and let stand until foamy, about 5 minutes. (If yeast doesn't foam, discard and start over with new yeast.)
Add 1/2 cup flour to yeast mixture and beat at medium speed until combined. Add whole eggs, yolk, vanilla, salt, and remaining 1/2 cup sugar and beat until combined. Reduce speed to low, then mix in remaining 2 3/4 cups flour, about 1/2 cup at a time. Increase speed to medium, then beat in butter, a few pieces at a time, and continue to beat until dough is shiny and forms strands from paddle to bowl, about 4 minutes. (Dough will be very soft and sticky.)
Scrape dough into a lightly oiled bowl and cover bowl with plastic wrap. Let dough rise in a draft-free place at warm room temperature until doubled in bulk, 1 1/2 to 2 hours.
Assemble babkas with filling:
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.
Make another babka with remaining dough, some of egg wash, and remaining butter and chocolate in same manner. Chill remaining egg wash, covered, to use later. Loosely cover pans with buttered 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. (Alternatively, let dough rise in pans in refrigerator 8 to 12 hours; bring to room temperature, 3 to 4 hours, before baking.)
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.