Preheat your oven to 375°F (190°C). Line a 12-by-17-inch rimmed baking sheet or jelly roll pan with parchment paper.
In a large bowl, use an electric mixer to beat unsalted butter with granulated white sugar at medium speed until light and fluffy, about 2 minutes.
Reduce the mixer speed to low and gradually beat in all-purpose flour, ground cinnamon, ground nutmeg, and kosher salt until a soft dough forms.
Press the dough evenly over the bottom of the prepared pan and ½ inch up the sides to form a crust.
Bake the crust in the center of the oven for about 15 minutes, or until golden and set. Let cool on a rack.
Meanwhile, in a very large, deep skillet, melt the remaining unsalted butter with light brown sugar.
Add the thinly sliced Granny Smith apples to the skillet and cook over high heat, stirring occasionally, until softened, about 10 minutes.
Stir in the ground cinnamon and ground nutmeg, cooking for an additional 10 minutes until caramelized and very tender.
Let the apple filling cool while you prepare the topping. In a large bowl, combine quick cooking oats, all-purpose flour, light brown sugar, ground cinnamon, baking soda, and kosher salt.
Using your fingers, squeeze the chilled unsalted butter into the oats and sugar mixture. Crumble it together until it resembles coarse meal.
Spread the apple filling evenly over the cooled crust. Scatter the crumble topping over the apples.
Bake the assembled bars for 50 to 60 minutes, or until the topping is golden brown. Let cool completely on a rack before cutting into 2-inch squares.