Heat the oven to 350°F.

Spread the bread cubes on an 18x13-inch rimmed baking sheet and bake for 10-15 minutes, until dry to the touch but not browned.

Butter a 9x13-inch baking dish with 1 tablespoon of the butter.

Melt the remaining 2 tablespoons butter in a 10-inch skillet over medium heat, add the apples, and cook for 5-7 minutes, stirring occasionally, until just beginning to soften.

In a large bowl, whisk the eggs, whole milk, brown sugar, cinnamon, salt, and 1/2 cup of the caramel sauce until combined.

Add the bread, cooked apples, and pecans, and fold until coated.

Let stand for 15 minutes, pressing the bread down occasionally so it absorbs the custard.

Transfer the mixture to the prepared baking dish and bake uncovered for 45-55 minutes, until golden on top and set in the center at 160°F.

Whisk the cornstarch and cold water in a small bowl until smooth.

Heat the heavy cream and granulated sugar in a small saucepan over medium heat until steaming, then whisk in the cornstarch mixture and cook for 1-2 minutes, whisking constantly, until slightly thickened.

Remove the saucepan from the heat and stir in the vanilla extract.

Let the pudding rest for 15 minutes after baking.

Microwave the remaining 1/4 cup caramel sauce in a microwave-safe bowl for 15 seconds, then in 10-second bursts until pourable.

Drizzle the caramel sauce over portions and serve with the warm vanilla sauce.