Heat the oven to 375°F. Grease an 8 by 8-inch square cake pan with cooking spray or softened butter. Line the pan with a long piece of parchment paper that extends up and over two opposite sides of the pan.

To make the crumb topping and crust by hand, whisk together the flour, granulated sugar, and salt in a large bowl. Rub the butter into the dry ingredients using your fingers, until the butter is crumbly and pea-sized. Stir in the egg and vanilla with a fork, or your hands, until the mixture holds together when squeezed.

Alternatively, to make the topping and crust with a food processor, pulse the dry ingredients and the butter in the bowl of the processor until the butter is pea-sized; then pulse in the egg and vanilla.

Remove 1½ cups (230 grams) of the crumb and refrigerate in a small bowl. Evenly press the remaining dough into the bottom of the prepared pan with floured fingers, dock with a fork, and bake for about 15 minutes.

To make the apple filling, whisk together the granulated sugar, flour, cinnamon, and salt in the same large bowl in which you made the crust and topping. Add the apples and toss to coat with a flexible spatula or your hands. Sprinkle with the lemon juice and toss a final time.

Remove the crust from the oven, if you have not already, and carefully cover the warm bottom crust with the filling, pressing it down lightly. Sprinkle with the remaining chilled dough, using your fingers to pinch off and create crumbs, and gently press to adhere.

Lower the oven temperature to 350°F and bake for 60 to 65 minutes, rotating at the halfway point, until the crumb topping is lightly browned and the apples bubble around the edges and are soft in the center.

Remove from the oven and let cool to room temperature. Serve straight from the pan, or lift the bars out of the pan by the parchment overhang, running a butter knife around the edges. Dust with confectioners’ sugar, slice, and serve. Keep the bars in an airtight container in the refrigerator for up to 3 days.