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MAKE THE CRUST: Preheat the oven to 375.
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Line a 15-by-17-inch rimmed baking sheet or jelly roll pan with parchment paper.
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In a standing electric mixer fitted with a paddle, beat the butter with the sugar at medium speed until light and fluffy, about 2 minutes.
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At low speed, beat in the flour and salt until a soft dough forms.
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Press the dough over the bottom of the prepared pan and 1/2 inch up the side in an even layer.
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Bake in the center of the oven for about 20 minutes, until the crust is golden and set.
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Let cool on a rack.
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MEANWHILE, MAKE THE FILLING: In each of 2 large skillets, melt 3 tablespoons of the butter with 1/4 cup of the light brown sugar.
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Add the apples to the skillets and cook over high heat, stirring occasionally, until softened, about 10 minutes.
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Stir half of the cinnamon and nutmeg into each skillet.
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Cook until the apples are caramelized and very tender and the liquid is evaporated, about 10 minutes longer; scrape up any bits stuck to the bottom of the skillets and add up to 1/2 cup of water to each pan to prevent scorching.
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Let cool.
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MAKE THE TOPPING: Spread the walnuts in a pie plate and toast until golden and fragrant, about 8 minutes.
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14
Let cool, then coarsely chop the walnuts.
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15
In a large bowl, mix the oats with the flour, light brown sugar, cinnamon, baking soda and salt.
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16
Using a pastry blender or two knives, cut in the butter until the mixture resembles coarse meal.
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Stir in the walnuts and press the mixture into clumps.
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Spread the apple filling over the crust.
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Scatter the crumbs on top, pressing them lightly into an even layer.
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Bake in the center of the oven for 1 hour, until the topping is golden; rotate the pan halfway through baking.
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Let cool completely on a rack before cutting into 2-inch bars.