Heat the oven to 250 degrees.
Line 2 baking sheets with parchment paper. Spread the doughnut pieces on the baking sheets and bake for about 30 minutes, until they are dry on the outside and semi-firm in the center. Set aside.
In a large bowl, whisk two cups of the cream with the milk, egg yolks, whole eggs and condensed milk. Add the doughnut pieces, and let them soak until soft, about 45 minutes.
Raise the temperature of the oven to 350 degrees. Lightly butter a 9x3-inch baking dish. Spoon or pour the doughnut mixture in the dish and cover with aluminum foil.
Set the dish in a roasting pan large enough to hold it, and add about ½ inch of water to the larger pan, creating a water bath.
Bake 40 minutes. Remove the foil and bake an additional 20 minutes, or until the custard is set. (Test by dipping a paring knife into the center.)
Heat the broiler. Being quite vigilant, broil the pudding for 3 minutes, or until the top is lightly browned. Let the pudding cool for about 30 minutes.
In a medium bowl, whip the remaining 2 cups of cream into semi-soft peaks. Stir in the espresso, and serve alongside the bread pudding.