Preheat the oven to 375°F. Line a 12-by-17-inch rimmed baking sheet or jelly roll pan with parchment paper.
With an electric mixer, beat the butter with the sugar at medium speed until light and fluffy, about 2 minutes.
At low speed, beat in the flour, spices, and salt until a soft dough forms.
Press the dough over the bottom of the prepared pan and 1/2 inch up the side in an even layer.
Bake in the center of the oven for about 15 minutes, or until the crust is golden and set. Let cool on a rack.
In a very large, deep skillet, melt the butter with the brown sugar. If you don't have a large skillet, divide between 2 skillets.
Add the apples to the skillet and cook over high heat, stirring occasionally, until softened, about 10 minutes.
Stir in cinnamon and nutmeg. Cook until the apples are caramelized and very tender and the liquid is evaporated, about 10 minutes longer.
Let cool.
In a large bowl, mix the oats with the flour, brown sugar, cinnamon, baking soda, and salt.
Using fingers (wash your hands and take off your rings first!), squeeze the butter into the oats and sugar and crumble together until the mixture resembles coarse meal.
Spread the apple filling over the crust. Scatter the crumbs on top, pressing them lightly into an even layer.
Bake in the center of the oven for 50 to 60 minutes, or until the topping is golden.
Let cool completely on a rack before cutting into 2-inch bars.