Make the dough: Sprinkle the yeast over the milk in a liquid measuring cup; add a pinch of sugar and set aside until bubbly, about 7 minutes. Combine the flour, the remaining 1/3 cup sugar, the eggs, yeast mixture, salt, vanilla, nutmeg and lemon zest in a large bowl. Stir with a wooden spoon to combine. Turn the dough out onto a work surface and knead until soft and smooth, about 5 minutes. Knead in the butter in three additions, dusting the dough with flour if it's too sticky. Transfer the dough to a large bowl; cover with plastic wrap and let rise at room temperature, about 1 1/2 hours. Punch down the dough, re-cover with plastic wrap and let rise in the fridge overnight.
Make the filling: Heat the sugar, heavy cream and salt in a saucepan until scalding. Pour over the bittersweet chocolate chips, butter and vanilla in a bowl. Whisk until smooth and shiny. Let cool to room temperature.
Make the topping: Whisk the flour, sugar, cocoa powder and salt in a separate bowl; work in the butter with your fingers until the mixture is sandy and chunky. Stir in the mini chocolate chips; set the topping aside.
Form the loaves: Cut the dough in half with a bench scraper or chef's knife. Using a rolling pin, roll each half into a 12-by-16-inch rectangle. Using an offset spatula, spread the filling on both dough rectangles, all the way to the edges. Starting from a long side, tightly roll each rectangle into a log. Wrap each log in plastic wrap and refrigerate 15 minutes. Unwrap the logs; cut each in half lengthwise with a bench scraper or chef's knife. Twist the halves together a few times, starting from the middle. Coat two 9-by-5-inch loaf pans with cooking spray and line with parchment, then spray the parchment. Place a dough twist snugly in each pan. Cover with plastic wrap and let rise 1 1/2 hours.
Finish and bake the babka: Preheat the oven to 350 degrees F. Brush each loaf with butter and sprinkle with the topping. Bake until browned, about 45 minutes. Meanwhile, make some simple syrup: Combine the sugar and water in a saucepan; simmer, stirring, until the sugar dissolves. Let cool. Pull the loaves out of the oven and immediately poke a bunch of holes in each with a wooden skewer. Pour 1 1/4 cups simple syrup evenly over the loaves.
Let sit 10 minutes, then remove the babka from the pans, remove the parchment and let cool completely on a rack.