Preheat oven to 375°F (190°C). Lightly grease an 8x4-inch loaf pan.
Whisk together the sugars, melted butter, vanilla, cinnamon, and salt. Whisk in the egg, then the milk and heavy cream.
Add the bread cubes and gently toss to coat. Let sit for 5–10 minutes, pressing the bread into the custard once or twice.
Transfer the bread and any remaining custard to the pan. Spread evenly and sprinkle with turbinado sugar.
Bake for 25–30 minutes, until golden and the center is set with a slight jiggle. Cool for 10 minutes.
For the sauce, combine the butter, brown sugar, heavy cream, and salt in a small saucepan. Cook over medium-low heat, stirring, until melted.
Increase heat to medium and bring to a boil. Remove from heat, stir in the vanilla, and cool for 10 minutes to thicken slightly. Drizzle over the bread pudding and serve.
