Preheat oven to 300°F. Place the challah cubes on a large rimmed baking sheet and toast until golden and crisp, about 30 minutes stirring halfway through. Set aside. Increase oven temperature to 325°F.

Grease a 2 quart baking dish (or two smaller casserole dishes) and set aside.

While the bread is toasting prepare the apple mixture. Heat a medium saucepan over medium heat and add 3 tablespoons of butter. Once melted, add the apples, granulated sugar and 2 tablespoons water. Cook, stirring frequently, until the apples are softened, about 15 minutes. Add the apple brandy and stir until the sauce is syrupy, about 1 minute. Remove from the heat.

Layer half the toasted bread cubes in the prepared pan. Using a slotted spoon, scoop out half the apples and spread evenly over the bread. Add the remaining bread to the pan, then top with the remaining drained apples. Reserve the apple juice and set aside.

In a large mixing bowl with a spout, whisk together the egg yolks, brown sugar, half & half, milk, cinnamon, nutmeg, vanilla and salt. Add the reserved apple juice and whisk again to combine.

Slowly pour the egg mixture over the bread and apples. Gently press down on the bread using the back of a spatula to ensure all pieces are moistened. Allow the mixture to soak for 15-20 minutes. Cover the pan with foil and bake for 45 minutes.

Remove the foil from the pudding, drizzle with 2 tablespoons melted butter, then sprinkle with 2 tablespoons coarse sugar. Return to the oven and bake, uncovered, for 20 to 25 minutes.

Remove from the oven and cool on a rack for 30 minutes (or more) before serving. Store leftovers in the refrigerator. Serve with Apple Brandy Sauce or ice cream if desired.

In a small mixing bowl, whisk together the egg yolks, sugar and salt until slightly thickened and lighter in color. Set aside.

In a medium saucepan, warm the milk over medium heat just until steaming but not boiling.

Slowly drizzle/pour half the warm milk into the yolk mixture while whisking continuously. This method of tempering the eggs will keep them from scrambling.

Pour the now warm yolk mixture back into the saucepan, whisking to combine. Cook over medium heat, stirring constantly with a wooden spoon, until the mixture thickens and coats the back of the spoon. This will only take a few minutes.

Place a medium mixing bowl over an ice bath (place the medium bowl inside a larger bowl filled with ice). Strain the custard through a fine-mesh sieve set over the bowl. Add the vanilla bean paste and stir to combine. Continue stirring the mixture until it cools. Add the apple brandy and refrigerate for at least 2 hours before serving.