Direction :

1. Preheat the oven to 325ºF. Butter a 9×13-inch baking dish and set aside. Arrange the brioche slices on a separate baking sheet. Place in the oven until lightly toasted, about 10 -15 minutes. Remove from the oven and set aside to cool.
2. Whisk the egg yolks and the 3/4 cup sugar until smooth. Add the milk, vanilla, and salt and whisk until fully incorporated. 
3. Tear the toasted bread slices into shards or cut into cubes and place in the prepared pan. Pour the custard evenly over the bread, filling the dish to the top. Let the mixture stand for 30 minutes, so that the bread can absorb the custard.
4. After 30 minutes, brush the top with the 2 tablespoons melted butter and sprinkle with the remaining 2 tablespoons of sugar. 
5. Bake the pudding for roughly 45 minutes. Let the pudding cool for about 10 minutes. 
6. If you’d like to serve it Tartine style, melt a tablespoon or two of butter in a large skillet over medium heat. Add your fruit of choice and cook for 3 to 5 minutes or until slightly softened. Add a few tablespoons of caramel sauce (recipe below) and swirl to coat. Spoon this caramel-y fruit sauce over slices of the warm bread pudding. 
7. Pour the cream into a small, heavy saucepan. Split the vanilla bean in half lengthwise and use the tip of a sharp knife to scrape the seeds from the pod halves into the cream. Place over medium-high heat and bring to just under a boil, stirring occasionally. Reduce the heat to low to keep the cream warm.
8. In a medium, heavy saucepan, combine the sugar, water, salt and corn syrup. Bring to a boil over medium heat, stirring to dissolve the sugar. Then cook, without stirring, until the mixture is amber colored, 5 to 8 minutes. Remove from the heat.
9. The mixture will continue to cook off the heat and become darker, so make sure to have your cream close by. Carefully and slowly add the cream to the sugar syrup. The mixture will boil vigorously at first. Let the mixture simmer down, and then whisk until smooth. Add the lemon juice. Let cool for about 10 minutes.
10. Cut the butter into 1-inch chunks and add them to the caramel one at a time, whisking constantly after each addition. Then whisk the caramel periodically as it continues to cool.