Adjust oven rack to middle position and heat oven to 300°F Toast bread on rimmed baking sheet, stirring occasionally, until golden and crisp, about 30 minutes. Transfer to large bowl.
Increase oven temperature to 325°F Grease 13- by 9-inch baking pan. Heat 1½ cups cream, milk, cocoa, espresso, and ½ cup sugar in saucepan over medium-high heat, stirring occasionally, until steaming and sugar dissolves. Pour warm cream mixture over toasted bread and let stand, tossing occasionally, until liquid has been absorbed, about 10 minutes.
Meanwhile, bring additional 1 cup cream to simmer in saucepan over medium-high heat. Remove from heat and stir in chocolate until smooth. Transfer 1 cup chocolate mixture to medium bowl and let cool 5 minutes (cover pan and reserve remaining chocolate mixture for serving). Add egg yolks, remaining cream, and remaining sugar to bowl with chocolate mixture and whisk to combine.
Transfer soaked bread mixture to prepared pan and pour chocolate custard mixture evenly over bread. Bake until pudding is just set and surface is slightly crisp, about 45 minutes. Let cool 30 minutes. Warm reserved chocolate mixture over low heat, then pour over bread pudding. Serve. (Leftover bread pudding should be refrigerated; reheat individual portions in microwave.).
Make Ahead: In step 4, once soaked bread mixture has been transferred to prepared pan, the pan can be covered with plastic wrap and refrigerated overnight. When ready to bake, remove plastic and proceed with recipe as directed, increasing baking time to 55 to 60 minutes. Let reserved chocolate serving sauce cool, then cover with plastic wrap and refrigerate. Heat sauce in microwave when needed.