Prepare the Oven and Baking Pan

Heat the oven to 375°F (190°C).

Line an 8×8-inch square baking pan with an 8-inch-wide strip of parchment paper, leaving enough excess paper extending over two opposite sides of the pan. The overhang will serve as handles for removing the cooled bars.

Prepare and Bake the Crust

Place the softened butter, granulated sugar, light brown sugar, and vanilla extract in a large mixing bowl. Using a stand mixer or handheld electric mixer, beat on medium speed for approximately 2 minutes, or until creamy and slightly lighter in appearance.

In a separate bowl, whisk together the flour, kosher salt, ground cinnamon, and finely chopped walnuts.

With the mixer running on low speed, gradually incorporate the dry ingredients into the butter mixture. Mix only until the ingredients come together into a dough.

Transfer approximately two-thirds of the dough to the prepared baking pan. Press it gently and evenly across the base, extending the dough about 1/2 inch up the sides of the pan. Pay particular attention to pressing the mixture into the corners so the finished bars have firm edges.

Set aside the remaining one-third of the dough for the topping.

Refrigerate the prepared crust for 20 minutes.

Bake the chilled crust for 18–20 minutes, or until it develops a golden-brown color. Remove it from the oven and set it aside to cool while preparing the filling.

Cook the Apple Filling

While the crust is chilling and baking, melt the 2 tablespoons of unsalted butter in a large, heavy pot over medium-low heat.

Add the sliced apples, lemon juice, granulated sugar, 1 1/2 teaspoons cinnamon, and freshly grated nutmeg.

Cook the apple mixture for approximately 15 minutes, stirring regularly, until the fruit becomes tender and most of the released liquid has evaporated.

During the final minute of cooking, stir in the cornstarch. Continue cooking until the juices thicken, then remove the pot from the heat.

Assemble and Bake

Spoon the cooked apple mixture evenly over the baked crust, keeping approximately a 1/2-inch border around the perimeter.

Use your fingers to break or press the reserved crust dough into irregular pieces, then distribute them evenly across the apple layer. The topping does not need to completely conceal the filling.

Return the pan to the 375°F (190°C) oven and bake for 25–30 minutes, or until the crumb topping is attractively browned.

Cool and Portion

Allow the apple pie bars to cool completely in the baking pan.

Once cooled, grasp the parchment-paper overhang and carefully lift the entire slab from the pan. Transfer it to a cutting board and divide it into individual bars.