Preheat the oven to 375°F (190°C) and generously grease a 9 x 13-inch baking pan to ensure easy release of the bars.
In a mixing bowl, blend 2 ½ cups of all-purpose flour with 2 tablespoons of granulated sugar. Incorporate 1 cup of butter or shortening using a pastry cutter until the mixture achieves a coarse, crumbly texture.
In a separate vessel, beat together 2 egg yolks and ½ cup of milk. Gradually stir this into the flour mixture, blending until a dough is formed.
Set aside one-quarter of the dough for the topping. Evenly press the remaining dough into the bottom of the prepared pan, forming a compact and level crust.
Distribute the thinly sliced Granny Smith apples uniformly across the crust. In a bowl, combine ¼ cup brown sugar, ½ cup granulated sugar, and 1 teaspoon ground cinnamon. Sprinkle this spiced sugar blend over the apple layer.
For the topping, mix the reserved dough with 1 tablespoon brown sugar, 1 tablespoon granulated sugar, 2 tablespoons flour, and a handful of crushed pecans. Stir until crumbly, then scatter over the apple layer.
Transfer to the oven and bake for 40 minutes, or until the top is golden and the apples are bubbling.
Remove from the oven and allow the bars to cool completely in the pan to ensure clean slices. Cut into squares and serve.