Lightly grease a 9x13-inch baking dish with butter or nonstick spray.
Add the cubed brioche to the prepared baking dish and spread it into an even layer.
In a large bowl, whisk together the pumpkin puree, eggs, egg yolks, melted butter, heavy cream, cinnamon, pumpkin pie spice, vanilla bean paste, kosher salt, brown sugar, and granulated sugar.
Whisk until the mixture is smooth and fully combined.
Slowly pour the pumpkin custard evenly over the brioche, making sure to cover as much of the bread as possible.
Gently press the bread down into the custard using the back of a spoon or your hands.
Let the bread pudding sit at room temperature for at least 1 hour (or cover overnight in the refrigerator).
During the first 10–15 minutes, gently press the bread down once or twice to help the brioche absorb the custard evenly.
Preheat the oven to 350°F.
Sprinkle the top evenly with turbinado sugar.
Loosely cover the baking dish with foil and bake for 25–30 minutes.
Remove the foil and continue baking for another 20–25 minutes, or until the top is golden brown and the center is just set but still soft and slightly custardy.
If the top is browning too quickly, loosely tent it with foil during the final portion of baking.
Remove from the oven and let rest for 10–15 minutes before serving.
Serve the bread pudding warm with a generous drizzle of warm caramel or toffee sauce.
Add a spoonful of freshly whipped cream and finish with a small sprinkle of flaky sea salt, if desired.