Preheat the oven to 350 degrees F and butter 6 (1 1/4 cup) ramekins or a shallow 2-quart ceramic or glass baking dish. Line the bottom of a baking pan large enough to snugly hold the ramekins or baking dish with a clean kitchen towel and bring a large saucepan of water to a boil. (The towel will prevent the ramekins from hitting the pan as they bake.)
Spread the bread out in a single layer on a baking sheet and place in the oven until slightly dried out, about 7 minutes. Let cool. Cut the bread into 1/2-inch cubes and divide evenly among the ramekins.
Combine the butter, sugar, and 2 tablespoons water in a large saucepan. Stir over low heat until smooth and bubbling at the edges, about 10 minutes. Add 2 cups of the half-and-half and the salt. Increase the heat to medium-high and bring almost to a boil.
Whisk the eggs and egg yolks with the remaining cup of half-and-half and the vanilla in a large bowl until well blended. Slowly pour in the hot half-and-half mixture, whisking constantly. Ladle the mixture evenly into the ramekins or baking dish.
Arrange the filled ramekins or baking dish on the towel in the pan. Place the pan on the middle rack of the oven and carefully pour in enough of the boiling water to reach about halfway up the sides of the dishes. Cook until golden and set in the middle, 35 to 40 minutes for ramekins, 40 to 45 minutes for the 2-quart dish. Transfer to wire racks to cool.
Prepare the Vanilla Sauce as directed. Stir the espresso powder into the sauce while warm. Cool and chill before serving with the pudding.
Combine the milk and vanilla bean in a small saucepan. Heat over medium-high heat until small bubbles form around the edge. Remove from the heat, cover, and infuse the milk for 10 minutes. Scrape the seeds of the bean into the milk. Rinse and dry the bean and save for another use (such as putting it in your bag of sugar, which will turn it into vanilla sugar).
Whisk the egg yolks with the sugar and salt in a large heatproof bowl until thick and light, 3 to 4 minutes. Stir in the hot milk. Return the custard to the pan and stir over low heat until slightly thickened or until a finger drawn across a wooden spoon dipped in the sauce leaves a mark (about 160 degrees F). Do not boil, or the sauce will curdle. Strain into a large bowl and cool to room temperature. Cover with plastic wrap and refrigerate until well chilled, at least 2 hours.
Recipe courtesy Sara Moulton, Sara Moulton Cooks at Home, Broadway Books, 2002