Preheat oven to 300 degrees F. Place the cubed challah on a rimmed baking sheet and toast in the oven until golden and crisp, 30 to 40 minutes. Transfer to a large bowl.
Increase the oven temperature to 325 degrees F. Grease a 9x13-inch baking dish.
In a medium saucepan, heat 1½ cups of the heavy cream, the milk, ½ cup of the sugar, the cocoa powder and espresso powder over medium-high heat until steaming and the sugar has dissolved, stirring occasionally. Pour the warm mixture over the toasted challah and let stand, tossing occasionally, until all of the liquid has been absorbed, about 10 minutes.
Meanwhile, bring 1 cup of the remaining heavy cream to a simmer in the now-empty saucepan over medium-high heat. Remove from the heat and stir in the chocolate until completely melted and smooth. Transfer 1 cup of the chocolate mixture to a medium bowl and allow to cool for 5 minutes (set aside the remaining chocolate mixture). Add the egg yolks, remaining 1½ cups heavy cream and remaining ½ cup sugar to the bowl with the chocolate mixture and whisk to combine.
Transfer the soaked challah mixture to the prepared pan and pour the chocolate custard mixture evenly over it. Bake until the pudding is just set and the surface is slightly crisp, about 45 minutes. Allow to cool for at least 30 minutes before serving. Warm the reserved chocolate mixture over low heat and pour over the pudding prior to serving. Leftover bread pudding should be covered in plastic wrap and refrigerated for up to 3 days.