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To make the crust, preheat the oven to 375 degrees F (190 C). Lightly spray a 9-by-12-inch baking pan with cooking spray
In a large bowl, using an electric mixer set on high speed, beat together the butter (1 cup) and granulated sugar until pale and fluffy, about 2 minutes.
Reduce the speed to low and beat in about ½ cup of the flour and the salt (1/4 teaspoon), then add the remaining flour (1/2 cup) gradually and beat until the dough comes together into a soft ball.
Transfer the dough to the prepared pan and, using your hands and a gentle touch, spread it out evenly over the bottom and lightly up the sides.
Bake until the top is golden brown, 15-20 minutes. Transfer to a wire rack and let cool completely. Leave the oven on.
To make the filling, place the butter (3 tablespoons) and brown sugar (1/4 cup) in a large frying pan. Place over low heat and, using a wooden spoon, stir to melt the sugar into the butter to for a syrup.
Add the apples, toss to coat with the syrup, and raise the heat to medium-high. Cook stirring often, until the apples have softened and are beginning to caramelize in the syrup, 15-20 minutes. If the pan begins to burn, sprinkle in a little water. Scrape the pan bottom, and reduce the heat.
When the apples are very soft, stir in the cinnamon and nutmeg. Remove from the heat and let cool completely.
To make the topping, in a large bowl, stir together the oats, flour (1/2 cup), brown sugar (1/2 cup), cinnamon (1/2 teaspoon), baking soda, and salt (1/4 teaspoon). Scatter pieces of butter (1/2 cup) on top and pinch into the oats mixture with your fingertips. Continue to pinch until the mixture comes together into uniform coarse, sandy clumps.
Stir in the walnuts and refrigerate until ready to use.
Spread the apple filling over the baked crust in an even layer. Sprinkle the topping over and press lightly. Bake until the topping is golden, about 20 minutes. Let cool completely on a wire rack, then cut into bars and serve.