Preheat the oven to 350°F (175°C). Grease a 9x13-inch baking dish with butter or non-stick spray.

In a large bowl, whisk together milk, heavy cream, eggs, granulated sugar, and vanilla extract until well combined.

Add the cubed bread to the milk mixture, pressing down gently to ensure all pieces are soaked. Let it sit for 10 minutes.

In a saucepan over medium heat, melt the butter. Add brown sugar, miso paste, water, and salt. Stir continuously until the mixture thickens into a caramel sauce (about 5 minutes).

Pour half of the miso caramel sauce over the soaked bread mixture, gently folding to combine. Transfer the mixture to the prepared baking dish.

Bake for 40-45 minutes, or until the top is golden brown and the pudding is set. Let it cool slightly before serving.

Drizzle the remaining miso caramel sauce over the bread pudding before serving. Enjoy warm or at room temperature.