Preheat the oven to 350°F (175°C). Lightly grease a 10 × 15-inch baking pan or line it with parchment paper, leaving some overhang for easy removal.

In a large mixing bowl, combine 1 1/2 cups of flour, 1 1/2 tablespoons of granulated sugar, baking powder, and salt. Add 1/2 cup of cold, cubed butter and work it into the dry ingredients using a pastry blender or fork until the mixture resembles coarse crumbs.

In a small bowl, whisk together the beaten egg yolks and water. Add the mixture to the flour mixture and stir until a soft dough forms.

Roll out the dough on a lightly floured surface to fit the bottom of the prepared baking pan. Carefully transfer the dough to the pan and press it into an even layer.

In a large bowl, toss the peeled, cored, and sliced apples with the lemon juice. Add 1 3/4 cups of granulated sugar, 2 tablespoons of flour, and 1/2 teaspoon of cinnamon. Gently mix until the apples are evenly coated.

Spread the apple filling evenly over the prepared crust. Cut the remaining 2 tablespoons of butter for the filling into small pieces and scatter them over the apples.

To prepare the crumb topping, combine 1 cup of flour, brown sugar, and 1 teaspoon of cinnamon in a medium bowl. Add 2/3 cup of cold, cubed butter and work it into the dry ingredients until coarse crumbs form.

Sprinkle the crumb topping evenly over the apple filling, covering the surface as evenly as possible.

Bake for approximately 60 minutes, or until the topping is golden brown and the apples are tender. If the topping begins to brown too quickly, loosely cover the pie with aluminum foil.

Remove the pie from the oven and allow it to cool for at least 30 minutes so the filling can settle. Cut into 16 squares and serve warm or at room temperature. Enjoy with vanilla ice cream, if desired.