How to Make It
In a medium bowl, whisk the flour with the sugar and salt. In a stand mixer fitted with the dough hook, combine the milk with the yeast and let stand until foamy, about 5 minutes. Add the egg and egg yolk and sprinkle the dry ingredients on top. Mix at low speed for 2 minutes. Scrape down the side of the bowl and mix at medium speed until all of the dry ingredients are incorporated and the dough is smooth, about 5 minutes. Add all of the butter at once and mix at low speed until it is fully incorporated and a tacky dough forms, about 3 minutes; scrape down the side of the bowl as needed during mixing. Cover the bowl with plastic wrap and let the dough stand at room temperature for 1 hour.
Line a large baking sheet with parchment paper and coat the paper generously with nonstick baking spray. Scrape the dough out onto the parchment paper and cut the dough in half. Pat each piece into a neat square. Cover with plastic wrap and refrigerate overnight.
In a large heatproof bowl set over a saucepan of simmering water, melt both chocolates with the butter, stirring occasionally, until smooth. Let cool to room temperature, then stir in the cookie crumbs and honey.
Coat two 9-by-4-inch loaf pans with nonstick baking spray and line with parchment paper, allowing 2 inches of overhang on each of the long sides. Roll out each square of dough to a 16-inch square. Using an offset spatula, spread all but 1/2 cup of the filling in an even layer over the dough squares to within 1/2 inch of the edges. Starting at the long edge nearest you, tightly roll up each dough square jelly roll–style into a tight log.
Using a sharp knife, cut the logs in half crosswise. Using an offset spatula, spread 1/4 cup of the reserved filling on the top and sides of 2 of the halves. Set the other halves on top in the opposite direction to form a cross. Twist to form spirals and transfer to the prepared pans. Cover the loaves with a towel and let stand in a warm place until doubled in bulk, about 2 hours.
Preheat the oven to 375°. Bake the loaves in the center of the oven for about 45 minutes, until puffed and well browned. Let cool slightly, then use the parchment paper to lift the babkas out of the pans and onto a rack set over a baking sheet. Discard the paper.
In a heatproof bowl set over a saucepan of simmering water, melt both chocolates with the butter; stir until smooth. Stir in the corn syrup. Spread the glaze on top of the warm babkas and let stand until set, about 30 minutes.